Algebra I

What is a variable?

A variable is a letter that stands for a number we do not know yet, or a number that can change. In algebra we use letters like x, y, or n so we can write general rules that work for many numbers at once. A number on its own, like 7 or -3, is called a constant because its value never changes. The whole power of algebra comes from this one move: instead of solving 100 similar arithmetic problems, we write one expression with a variable and solve them all at once.

An algebraic expression is a combination of numbers, variables, and operations (add, subtract, multiply, divide). Examples: 3x + 5, 2n - 7, and 4(a + b). Notice that 3x means "3 times x" - in algebra we usually drop the multiplication sign between a number and a variable, because the symbol for multiplication looks too much like the letter x. When two variables are multiplied we simply write them side by side, so ab means "a times b".

Parts of an expression

A term is a single number, a single variable, or numbers and variables multiplied together. Terms are separated by + or - signs. In the expression 3x + 5 there are two terms: 3x and 5. The number multiplied by the variable is the coefficient; here the coefficient of x is 3. The lone number 5 is a constant term. When a variable appears with no number in front, like x by itself, its coefficient is understood to be 1, because x means 1x. Likewise -x has coefficient -1.

Learning to read an expression into its parts is like learning to read a sentence into subject and verb. Once you can point at the coefficient, the variable, and the constant of every term, the rules that come later have something concrete to grab onto.

Translating words into algebra

Word problems hide operations inside ordinary language. These words are your clues:

• Addition: sum, plus, more than, increased by, total, combined.
• Subtraction: difference, minus, less than, decreased by, fewer, reduced by.
• Multiplication: product, times, of, twice (times 2), doubled, tripled.
• Division: quotient, divided by, per, ratio, split equally.

Be careful with "less than": "5 less than a number" means x - 5, not 5 - x. The order flips, because you start with the number and take 5 away from it. The same reversal happens with "subtracted from": "3 subtracted from y" is y - 3. Reading these slowly and asking "what do I start with?" prevents most translation errors.

Worked example 1: write the expression

"Seven more than twice a number." Twice a number is 2x. Seven more than that adds 7: the expression is 2x + 7. Now a two-step one: "the product of 4 and the quantity three less than a number." The quantity "three less than a number" is (x - 3). The product of 4 and that quantity is 4(x - 3). Keeping the parentheses is essential, because 4 multiplies the entire difference, not just the x.

Worked example 2: evaluate an expression

To evaluate an expression means to find its value by substituting a number for the variable. Evaluate 3x² - 2x + 5 when x = -2. Substitute -2 everywhere x appears, using parentheses so the signs stay correct:

1. Replace x: 3(-2)² - 2(-2) + 5.
2. Exponent first: (-2)² = 4, so 3(4) - 2(-2) + 5.
3. Multiply: 12 + 4 + 5.
4. Add: 21.

The key habit is to substitute with parentheses and follow the order of operations, which you will study in depth in the next lesson. If you write 3 × -2² without parentheses you risk squaring only the 2 and losing the sign.

Worked example 3: a formula from real life

A phone plan charges 20 dollars per month plus 5 dollars for each gigabyte of data. If g is the number of gigabytes, the monthly cost is 20 + 5g. Evaluate it for a month with 3 gigabytes: 20 + 5(3) = 20 + 15 = 35 dollars. Notice how one expression handles every possible month; that is exactly why algebra is worth learning.

Why this matters

Every spreadsheet formula, every line of computer code that computes a total, and every scientific law is an algebraic expression in disguise. The skill of naming an unknown with a letter, writing a relationship, and then substituting values is the single most transferable idea in this course. Master it here and the rest of algebra is a series of new relationships to write and solve.

Worked example 4: two variables at once

Evaluate 2ab - 3b + 4 when a = 3 and b = -2. Substitute both values, using parentheses around each one:

1. Replace the variables: 2(3)(-2) - 3(-2) + 4.
2. Multiply the first term left to right: 2(3) = 6, then 6(-2) = -12.
3. Multiply the second term: -3(-2) = +6, because a negative times a negative is positive.
4. Add left to right: -12 + 6 = -6, then -6 + 4 = -2.

The value is -2. Nothing new happened here: you substituted twice and followed the same order of operations. Formulas in science class work exactly this way, with several inputs feeding one expression.

Real-world applications

Expressions are the working language of everyday calculation:

• Temperature conversion: the formula F = (9/5)C + 32 converts Celsius to Fahrenheit. At C = 25: (9/5)(25) = 45, and 45 + 32 = 77 degrees Fahrenheit.
• Fundraising: a club selling shirts for 12 dollars each after paying 150 dollars for supplies earns 12n - 150. Selling 40 shirts gives 12(40) - 150 = 480 - 150 = 330 dollars of profit.
• Data plans: the monthly cost 20 + 5g from the earlier example handles every possible month with one rule.
• Spreadsheets: a formula such as =B2*12-150 is literally the expression 12n - 150 with a cell reference playing the role of the variable.

Common misconceptions

• Believing 2x means 2 + x. A number written next to a variable always means multiplication, so when x = 7, 2x is 14, not 9.
• Translating "5 less than x" as 5 - x. It is x - 5: you start with x and remove 5 from it.
• Substituting a negative without parentheses. For x = -2, the term x² must become (-2)2 = 4; writing -2² gives -4, the wrong sign.
• Thinking a bare x has coefficient 0. The coefficient of x is 1, and the coefficient of -x is -1.
• Confusing x² with 2x. The first means x times x; the second means x plus x. At x = 3 they are 9 and 6, which are different numbers.

Recap

• A variable is a letter that stands for a number; a constant is a fixed number.
• Terms are separated by + and - signs, and the coefficient is the number multiplied by the variable.
• Translate word phrases carefully: "less than" and "subtracted from" reverse the order.
• To evaluate, substitute with parentheses and follow the order of operations.
• One expression can model every case of a real situation, which is the whole point of algebra.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 1: Foundations (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Algebra foundations" unit: introduction to variables and evaluating expressions (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra Preliminaries (tutorial.math.lamar.edu).

Why we need an agreed order

The expression 2 + 3 × 4 could be 20 if you add first, or 14 if you multiply first. To make sure everyone on Earth gets the same answer, mathematicians agree on one order of operations, often remembered as PEMDAS:

1. Parentheses (and other grouping symbols) first.
2. Exponents next.
3. Multiplication and Division, left to right (they share a level).
4. Addition and Subtraction, left to right (they share a level).

So 2 + 3 × 4 = 2 + 12 = 14. Multiplication is done before addition. The two "left to right" rules matter: 12 ÷ 4 × 3 is 3 × 3 = 9, not 12 ÷ 12 = 1, because you take division and multiplication in reading order. Thinking of MD as one shared level, and AS as one shared level, prevents the classic mistake of always doing multiplication before division.

Worked example 1: a numeric expression

Simplify 4 + 2(5 - 1)² ÷ 8.

1. Parentheses: 5 - 1 = 4, giving 4 + 2(4)² ÷ 8.
2. Exponent: 4² = 16, giving 4 + 2(16) ÷ 8.
3. Multiply and divide left to right: 2 × 16 = 32, then 32 ÷ 8 = 4, giving 4 + 4.
4. Add: 8.

The distributive property

The distributive property says a(b + c) = ab + ac. You multiply the outside number by each term inside the parentheses. This is how we remove parentheses in algebra, and it is one of the most used rules in the entire subject. Geometrically it says the area of a rectangle that is a tall and b + c wide equals the two smaller areas ab and ac added together.

Example: -2(3x - 4) = (-2)(3x) + (-2)(-4) = -6x + 8. Watch the signs carefully: a negative times a negative is positive, so the last term becomes +8. A very common slip is to distribute to the first term but forget the second, so make it a habit to draw two arrows from the outside factor to each inside term.

Like terms

Like terms have exactly the same variable part. 4x and 7x are like terms; 4x and 4x² are not, because one has an exponent. Constants like 7 and -3 are like terms with each other. To combine like terms, add or subtract their coefficients and keep the variable part the same. You are really just using the distributive property backwards: 4x + 7x = (4 + 7)x = 11x.

Worked example 2: simplify an algebraic expression

Simplify 4x + 7 - 2x - 3.

1. Group like terms: the x-terms are 4x and -2x; the constants are 7 and -3.
2. Combine the x-terms: 4x - 2x = 2x.
3. Combine the constants: 7 - 3 = 4.
4. Result: 2x + 4.

A slightly harder one: simplify 3(x + 2) - x. Distribute first: 3x + 6 - x. Then combine: 3x - x = 2x, so the answer is 2x + 6.

Worked example 3: distribute a negative, then combine

Simplify 5 - 2(3x - 4) + x. First distribute the -2 across the parentheses: 5 - 6x + 8 + x. Now gather the x-terms -6x + x = -5x and the constants 5 + 8 = 13. The simplified expression is -5x + 13. Notice the sign of every distributed term was tracked carefully; that discipline is what separates a right answer from a near miss.

Why this matters

Simplifying is the tidying step that makes every future problem shorter. Before you solve an equation, before you graph a function, before you factor, you almost always simplify each side first. Calculators and computers follow this exact order too, so understanding PEMDAS also means understanding why a spreadsheet formula returns the number it does.

Worked example 4: everything at once

Simplify 18 ÷ 3 + 2(8 - 5)2, showing every level of PEMDAS in order:

1. Parentheses: 8 - 5 = 3, giving 18 ÷ 3 + 2(3)2.
2. Exponent: 32 = 9, giving 18 ÷ 3 + 2(9).
3. Multiplication and division, left to right: 18 ÷ 3 = 6, then 2(9) = 18, giving 6 + 18.
4. Add: 24.

Every step resolved exactly one level of the hierarchy. Writing each new line completely, instead of doing two steps in your head, is the habit that keeps long simplifications error free.

Worked example 5: distribute twice, then combine

Simplify 2(3x + 1) + 4(x - 2).

1. Distribute the 2: 6x + 2.
2. Distribute the 4: 4x - 8.
3. Put the pieces together: 6x + 2 + 4x - 8.
4. Combine x-terms: 6x + 4x = 10x. Combine constants: 2 - 8 = -6.

The simplified expression is 10x - 6. You can spot-check with a number: at x = 1 the original is 2(4) + 4(-1) = 8 - 4 = 4, and the simplified form gives 10(1) - 6 = 4. They match, which is strong evidence the simplification is right.

Real-world applications

• Shopping totals: 3 shirts at 12 dollars and 2 hats at 9 dollars cost 3(12) + 2(9) = 36 + 18 = 54 dollars. Multiplication before addition is exactly what a cash register does.
• Sales tax: a 40 dollar item with 8 percent tax costs 40(1 + 0.08) = 40(1.08) = 43.20 dollars; the parentheses group the tax rate with the 1 before multiplying.
• Spreadsheets and code: every programming language evaluates 2 + 3 * 4 as 14. Programmers add parentheses to force any other order, exactly as you do on paper.
• Recipes and materials: doubling a recipe that needs 2 cups of flour and 3 cups of oats per batch means 2(2 + 3) = 10 cups total, a direct use of the distributive property.

Common misconceptions

• Always multiplying before dividing. Multiplication and division share one level and go left to right: 12 ÷ 2 × 3 = 18, not 2.
• Distributing to the first term only. In -2(3x - 4), the -2 must hit both terms: -6x + 8, not -6x - 4.
• Combining unlike terms. 4x + 3 cannot become 7x; a term with x and a plain number measure different things.
• Treating x and x² as like terms. 4x + 4x2 does not combine; the variable parts differ.
• Dropping parentheses around a squared negative. (-3)2 = 9, while -32 = -9; the parentheses decide whether the sign gets squared.

Recap

• PEMDAS orders operations: grouping symbols, exponents, multiply/divide left to right, add/subtract left to right.
• The distributive property a(b + c) = ab + ac removes parentheses; send the outside factor to every inside term.
• Like terms have identical variable parts; combine them by adding coefficients.
• Check a simplification by substituting an easy number into both forms.
• Simplify each side fully before you try to solve any equation.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 1: Foundations, sections on the language of algebra and properties of real numbers (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Algebra foundations" unit: order of operations and combining like terms (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra Preliminaries (tutorial.math.lamar.edu).

Families of numbers

The numbers you use in algebra belong to nested families, each one built by adding something new to the family before it:

• Natural numbers: 1, 2, 3, ... (the counting numbers).
• Whole numbers: the naturals plus 0.
• Integers: whole numbers and their negatives: ... -2, -1, 0, 1, 2, ...
• Rational numbers: any number that can be written as a fraction of two integers, such as 1/2, -3, or 0.25. Their decimals either stop or repeat.
• Irrational numbers: numbers whose decimals never stop and never repeat, such as the square root of 2 or the number pi.

Together, the rational and irrational numbers make up the real numbers - every point on the number line. The families are nested: every natural number is also a whole number, an integer, a rational number, and a real number. A number can belong to several families at once, so the useful question is usually "what is the smallest family this number belongs to?"

The number line and absolute value

Every real number has a home on the number line. Numbers to the right are larger; numbers to the left are smaller. The absolute value of a number is its distance from zero, so it is never negative. We write it with bars: |-4| = 4 and |4| = 4. Distance has no direction, which is exactly why absolute value throws away the sign.

Rules for signed numbers

These four rules cover almost everything:

• Adding same signs: add the values, keep the sign. -4 + (-3) = -7.
• Adding different signs: subtract the smaller value from the larger, keep the sign of the larger. -8 + 3 = -5.
• Multiplying or dividing: same signs give a positive; different signs give a negative. (-6)(-2) = 12 but (-6)(2) = -12.
• Subtracting: add the opposite. 5 - (-3) = 5 + 3 = 8.

A helpful mental picture for addition: think of positives as steps right and negatives as steps left on the number line. -8 + 3 means start at -8 and take 3 steps right, landing at -5. For multiplication, the sign rule comes from patterns: multiplying by a negative reverses direction, and reversing twice returns you to the original direction, which is why two negatives make a positive.

Worked example 1: a chain of signed operations

Compute -3 - (-7) + (-2)(4). Handle the multiplication first: (-2)(4) = -8. Rewrite the subtraction of a negative as addition: -3 + 7 + (-8). Now left to right: -3 + 7 = 4, then 4 + (-8) = -4. The result is -4.

Worked example 2: fractions with signs

Compute -3/4 + 1/6. The least common denominator of 4 and 6 is 12. Rewrite each fraction: -3/4 = -9/12 and 1/6 = 2/12. Now add the numerators over 12: -9/12 + 2/12 = -7/12. The answer is -7/12. When denominators differ you must rename before you add; you can never add numerators over different denominators.

Why this matters

Signed numbers describe temperature below zero, money owed, elevation below sea level, and electric charge. Getting the sign rules automatic frees your attention for the harder ideas ahead, because almost every equation you solve will involve a negative somewhere. Absolute value, meanwhile, reappears whenever a problem cares about size regardless of direction, such as measuring error or distance.

Worked example 3: classify a list of numbers

Name the smallest family for each number: -7, 0, 3/5, the square root of 9, the square root of 7, and 0.454545... (repeating).

1. -7 is negative, so it is not natural or whole; it is an integer (and also rational and real).
2. 0 is a whole number: the whole numbers are exactly the naturals plus 0.
3. 3/5 is a fraction of integers, so it is rational.
4. The square root of 9 simplifies to 3, so despite its costume it is a natural number. Always simplify before you classify.
5. The square root of 7 is not a perfect-square root; its decimal never stops or repeats, so it is irrational.
6. 0.454545... repeats the block 45 forever, so it is rational; in fact it equals 45/99, which reduces to 5/11.

Worked example 4: order signed numbers

Order from least to greatest: -3/2, -2, 0.75, -0.5, 1. Convert to decimals to compare: -3/2 is -1.5. On the number line, more negative means further left, so the order is -2 < -1.5 < -0.5 < 0.75 < 1. The most common error is thinking -2 is greater than -1.5 because 2 is greater than 1.5; with negatives, larger size means smaller value.

Real-world applications

• Temperature: if the temperature climbs from -8 degrees to 5 degrees, the change is 5 - (-8) = 5 + 8 = 13 degrees.
• Money: a balance of 40 dollars after a 55 dollar purchase becomes 40 - 55 = -15 dollars, an overdraft the negative sign makes visible.
• Elevation: Death Valley sits at about -282 feet and Denali's summit at about 20,310 feet; the vertical difference is 20310 - (-282) = 20592 feet.
• Error and tolerance: a machinist who cuts a rod 0.2 mm too long or 0.2 mm too short is off by |0.2| either way; absolute value measures the size of an error without caring about its direction.

Common misconceptions

• Thinking -x is always negative. The expression -x means "the opposite of x." If x = -3, then -x = 3, a positive number.
• Applying the multiplication sign rule to addition. Two negatives multiply to a positive, but -3 + (-4) = -7: adding two negatives goes further negative.
• Believing every unending decimal is irrational. Repeating decimals such as 0.333... are rational; only never-repeating, never-ending decimals are irrational.
• Treating 22/7 as pi. The fraction 22/7 is a rational approximation; pi itself is irrational and equals no fraction exactly.
• Ranking negatives by size. -100 is less than -1, even though 100 is bigger than 1; direction on the number line decides order.

Recap

• The number families nest: naturals inside wholes inside integers inside rationals, and rationals plus irrationals make the reals.
• Rational numbers have decimals that stop or repeat; irrational numbers never do either.
• Absolute value is distance from zero and is never negative.
• Same signs add and keep the sign; different signs subtract and keep the sign of the larger size; subtraction means adding the opposite.
• Multiplying or dividing two numbers with the same sign gives a positive, with different signs a negative.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 1: Foundations, sections on integers, fractions, decimals, and the real numbers (openstax.org/details/books/elementary-algebra-2e).
• OpenStax, Prealgebra 2e, Chapter 3: Integers (openstax.org/details/books/prealgebra-2e).
• Khan Academy, Algebra 1, "Irrational numbers" unit (khanacademy.org/math/algebra).

What it means to solve an equation

An equation says two expressions are equal, and it always contains an equals sign, like x + 5 = 12. To solve it means to find the value of the variable that makes the statement true. That value is the solution. An expression can only be simplified; an equation can be solved, because the equals sign gives you something to work toward.

The golden rule: an equation is like a balanced scale. Whatever you do to one side, you must do to the other side, or the balance breaks. We isolate the variable by undoing operations with their inverse operations: addition undoes subtraction, and multiplication undoes division. The goal of every step is to get the variable alone on one side.

Worked example 1: a one-step equation

Solve x + 5 = 12. The 5 is added to x, so we subtract 5 from both sides:

1. x + 5 - 5 = 12 - 5
2. x = 7

Check: substitute back: 7 + 5 = 12. True, so x = 7 is correct. Checking is not optional busywork; it is how you catch a sign slip before it costs you.

Worked example 2: a one-step with multiplication

Solve 3x = 21. The x is multiplied by 3, so divide both sides by 3: x = 7. Check: 3(7) = 21. Correct. A one-step equation involving multiplication is undone by division, and one involving division is undone by multiplication.

Worked example 3: a two-step equation

Solve 2x + 3 = 11. A good strategy is to undo addition and subtraction first, then undo multiplication and division. This reverses the order of operations, like taking off your shoes before your socks.

1. Subtract 3 from both sides: 2x = 8.
2. Divide both sides by 2: x = 4.

Check: 2(4) + 3 = 8 + 3 = 11. True.

Worked example 4: a fraction coefficient

Solve (2/3)x - 1 = 5.

1. Add 1 to both sides: (2/3)x = 6.
2. Multiply both sides by the reciprocal 3/2: x = 6 × 3/2 = 9.

Check: (2/3)(9) - 1 = 6 - 1 = 5. True. Multiplying by the reciprocal is the clean way to undo a fraction coefficient, because a number times its reciprocal is 1.

Worked example 5: the variable on the right

Solve 20 = 4 + 8x. The variable being on the right changes nothing about the method. Subtract 4 from both sides: 16 = 8x. Divide by 8: 2 = x, which we read as x = 2. Equations are symmetric, so x = 2 and 2 = x mean the same thing.

Why this matters and the pattern to remember

Notice the pattern in every problem: identify what is being done to the variable, then apply the inverse operation to both sides, and always check by substituting your answer back into the original equation. This single loop, "undo, keep balanced, check," scales up to the longest equations in the course. Solving equations is also the engine behind real questions: how many hours until you can afford something, what price hits a target profit, how far a car has traveled. Each of those becomes an equation you isolate the same way.

Worked example 6: a real two-step problem

A gym charges 25 dollars to join plus 15 dollars per month. You have paid 115 dollars in total. How many months have you been a member? Let m be the months.

1. Write the equation from the story: 25 + 15m = 115.
2. Subtract the joining fee from both sides: 15m = 115 - 25 = 90.
3. Divide both sides by 15: m = 90 / 15 = 6.

Check: 25 + 15(6) = 25 + 90 = 115. True, so you have been a member for 6 months. Notice the anatomy: the one-time fee is the constant, the monthly rate is the coefficient, and the total is the other side of the balance.

Real-world applications

• Saving for a goal: you have 80 dollars and save 30 per week; a 320 dollar phone means 80 + 30w = 320. Subtract 80: 30w = 240, so w = 8 weeks. Check: 80 + 240 = 320.
• Temperature conversion: if F = 1.8C + 32 and the reading is 86 degrees Fahrenheit, then 1.8C = 86 - 32 = 54, so C = 54 / 1.8 = 30 degrees Celsius. Check: 1.8(30) + 32 = 54 + 32 = 86.
• Splitting a cost: four friends share a pizza order plus a 6 dollar tip for a 38 dollar total: 4p + 6 = 38 gives 4p = 32, so each pizza share is p = 8 dollars. Check: 4(8) + 6 = 38.

Common misconceptions

• Undoing operations in PEMDAS order instead of reverse. In 2x + 3 = 11, remove the +3 first, then divide by 2; solving reverses the order of operations.
• Operating on one side only. Subtracting 5 from the left side but not the right breaks the balance and changes the solution.
• Stopping at -x = 5. That is not solved yet; multiply both sides by -1 to get x = -5.
• Dividing only part of a side. To solve 3x = 21 + 6, simplify the right side to 27 first, then divide both sides by 3 to get x = 9. If you divide before simplifying, every term must be divided, not just one.
• Skipping the check. Substituting the answer back takes seconds and catches nearly every sign or arithmetic slip.

Recap

• An equation states two expressions are equal; solving finds the value that makes it true.
• Use inverse operations: undo addition with subtraction, multiplication with division, and vice versa.
• Whatever you do to one side, do identically to the other side.
• For two-step equations, clear the constant first, then the coefficient.
• Undo a fraction coefficient by multiplying by its reciprocal, and always check by substitution.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 2: Solving Linear Equations and Inequalities (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Solving equations and inequalities" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Linear Equations" (tutorial.math.lamar.edu).

A general strategy

Longer equations follow a reliable recipe. Memorize the order and most equations become routine:

1. Remove parentheses using the distributive property.
2. Combine like terms on each side.
3. Get all variable terms on one side and all constants on the other, using addition or subtraction.
4. Divide to isolate the variable.
5. Check your answer.

The reason this order works is that it clears clutter before it collects: parentheses and like terms are handled while each side still stands alone, and only then do you move terms across the equals sign.

Worked example 1: distribute first

Solve 3(x - 4) = 2x + 5.

1. Distribute the 3: 3x - 12 = 2x + 5.
2. Subtract 2x from both sides: x - 12 = 5.
3. Add 12 to both sides: x = 17.

Check: left side 3(17 - 4) = 3(13) = 39; right side 2(17) + 5 = 39. Both equal 39, so it works.

Worked example 2: variables on both sides

Solve 5 - 2x = 3x - 10. When the variable appears on both sides, move it to whichever side keeps it positive to reduce sign errors.

1. Add 2x to both sides to collect x on the right: 5 = 5x - 10.
2. Add 10 to both sides: 15 = 5x.
3. Divide by 5: x = 3.

Check: left 5 - 2(3) = -1; right 3(3) - 10 = -1. Both are -1.

Special cases

Sometimes the variable disappears while you are solving. Two things can happen:

• No solution: if you end with a false statement like 7 = 3, the equation is never true. Example: 2x + 1 = 2x + 4 reduces to 1 = 4, which is impossible. Graphically these are two parallel lines that never cross.
• Infinitely many solutions: if you end with a true statement like 6 = 6, every number works. Example: 3(x + 2) = 3x + 6 reduces to 6 = 6. This is called an identity, and graphically it is the same line drawn twice.

Clearing fractions

When an equation has fractions, you can multiply every term by the least common denominator to clear them. For x/2 + 1 = x/3 + 3, multiply every term by 6: 3x + 6 = 2x + 18. Then x = 12. Clearing fractions early keeps the arithmetic clean, and it is one of the most useful habits for messy equations.

Worked example 3: distribute, combine, then solve

Solve 4(x + 1) - 2 = 2(3x - 1). Distribute both sides: 4x + 4 - 2 = 6x - 2. Combine on the left: 4x + 2 = 6x - 2. Subtract 4x: 2 = 2x - 2. Add 2: 4 = 2x. Divide: x = 2. Check: left 4(3) - 2 = 10; right 2(5) = 10. Both are 10, so x = 2.

Why this matters

Multi-step equations model real trade-offs, such as "at how many minutes do two phone plans cost the same?" That break-even question sets one cost expression equal to another, with the variable on both sides, and solving it tells you which plan wins in which range. Recognizing no-solution and identity cases also matters: they tell you when a situation can never happen, or when a claimed rule is true for every value.

Worked example 4: the break-even problem, fully worked

Plan A costs 10 dollars a month plus 0.25 per minute; Plan B costs 25 dollars a month plus 0.10 per minute. For how many minutes m do they cost the same?

1. Set the two cost expressions equal: 10 + 0.25m = 25 + 0.10m.
2. Subtract 0.10m from both sides: 10 + 0.15m = 25.
3. Subtract 10 from both sides: 0.15m = 15.
4. Divide by 0.15: m = 15 / 0.15 = 100 minutes.

Check: Plan A at 100 minutes: 10 + 0.25(100) = 10 + 25 = 35. Plan B: 25 + 0.10(100) = 25 + 10 = 35. Both cost 35 dollars, so 100 minutes is the break-even point. Below it Plan A is cheaper; above it Plan B wins.

Worked example 5: fractions on both sides

Solve (x + 2)/3 = (2x - 1)/5. The least common denominator of 3 and 5 is 15, so multiply both sides by 15:

1. Left side: 15(x + 2)/3 = 5(x + 2) = 5x + 10. Right side: 15(2x - 1)/5 = 3(2x - 1) = 6x - 3.
2. The equation is now 5x + 10 = 6x - 3.
3. Subtract 5x from both sides: 10 = x - 3.
4. Add 3 to both sides: x = 13.

Check: left side (13 + 2)/3 = 15/3 = 5; right side (2(13) - 1)/5 = 25/5 = 5. Both sides equal 5, so x = 13.

Real-world applications

• Geometry with a twist: a rectangle's perimeter is 54 meters and its length is twice its width: 2(2w) + 2w = 54, so 6w = 54 and w = 9; the garden is 9 by 18 meters. Check: 2(18) + 2(9) = 36 + 18 = 54.
• Comparing offers: break-even questions (phone plans, gym memberships, ride services, printing companies) all set one multi-step expression equal to another.
• Consistency checks: an engineer whose constraints reduce to a false statement knows the design is impossible; a reduction to a true statement means any value works and another requirement is needed.

Common misconceptions

• Losing a sign while distributing. -(2x - 7) becomes -2x + 7; both terms flip, not just the first.
• Moving a term without changing its sign. "Moving" 3x across the equals sign really means subtracting 3x from both sides, so it must reappear with the opposite sign.
• Multiplying only the fraction terms by the LCD. Every term on both sides gets multiplied, including whole numbers.
• Treating a vanished variable as a mistake. A false leftover like 1 = 4 means no solution; a true leftover like 6 = 6 means infinitely many. Both are real answers.
• Declaring victory at 2x = 18. The final step, dividing to get x = 9, is still required, and so is the check.

Recap

• Follow the recipe: distribute, combine like terms on each side, collect variables on one side and constants on the other, divide, check.
• Move the variable to the side that keeps its coefficient positive to avoid sign errors.
• Clear fractions early by multiplying every term by the least common denominator.
• A false ending statement means no solution; a true one means infinitely many (an identity).
• Break-even problems are equations with the variable on both sides.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 2: Solving Linear Equations and Inequalities, sections on variables on both sides and a general solving strategy (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Solving equations and inequalities" unit: equations with variables on both sides (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Linear Equations" (tutorial.math.lamar.edu).

Ratios and proportions

A ratio compares two quantities, such as 3 cups of flour to 4 cups of water, written 3/4 or 3 to 4. A ratio keeps its meaning when both parts are scaled by the same factor, so 3 to 4 is the same relationship as 6 to 8 or 30 to 40. A proportion is an equation stating that two ratios are equal, such as 3/4 = x/20.

To solve a proportion, use cross multiplication: for a/b = c/d, the cross products are equal, so ad = bc. This works because multiplying both sides of the proportion by both denominators clears the fractions in one move.

Worked example 1: solve a proportion

Solve 3/4 = x/20.

1. Cross multiply: 4 · x = 3 · 20, so 4x = 60.
2. Divide by 4: x = 15.

Check: 15/20 reduces to 3/4. Correct.

Worked example 2: a real-world rate

If a car travels 150 miles on 5 gallons of gas, how far can it travel on 8 gallons at the same rate? Set up a proportion of miles to gallons, keeping the same quantity on top in both ratios: 150/5 = m/8. Cross multiply: 5m = 1200, so m = 240 miles. The most common setup error is flipping one ratio; guard against it by labeling units on top and bottom.

Worked example 3: scale and maps

On a map, 2 centimeters represents 15 kilometers. How many kilometers does 9 centimeters represent? Write 2/15 = 9/d, keeping centimeters over kilometers on both sides. Cross multiply: 2d = 135, so d = 67.5 kilometers. Scale drawings, model kits, and blueprints are all proportion problems.

Formulas and literal equations

A formula is an equation that relates several quantities, like d = rt (distance equals rate times time) or the perimeter of a rectangle P = 2L + 2W. A literal equation contains several letters, and sometimes we need to solve for one letter in terms of the others. You use the very same balancing moves as before; the only difference is that your answer will contain letters instead of a single number.

Worked example 4: rearrange a formula

The formula d = rt gives distance. Solve it for time t. Since r multiplies t, divide both sides by r:

t = d/r.

Worked example 5: solve for W

Solve P = 2L + 2W for W.

1. Subtract 2L from both sides: P - 2L = 2W.
2. Divide both sides by 2: W = (P - 2L)/2.

Rearranging formulas is a powerful skill: it lets you reuse one relationship to find whichever quantity you need. Scientists rearrange the same formula constantly, solving d = rt for r to get speed, or for t to get travel time, without ever memorizing three separate formulas.

Why this matters

Proportions power unit conversion, recipe scaling, currency exchange, medicine dosing by body weight, and reading any map or scale model. Rearranging formulas is the daily work of every science and engineering field, where one law such as Ohm's law or the area of a circle must be solved for whichever variable is unknown. Both skills come down to the balancing you already know, applied with letters in the mix.

Worked example 6: unit price decides the better buy

A 12 ounce jar of peanut butter costs 3.60 dollars and a 20 ounce jar costs 5.40 dollars. Which is the better deal? Compute each unit price, the ratio of dollars to ounces:

1. Small jar: 3.60 / 12 = 0.30 dollars per ounce.
2. Large jar: 5.40 / 20 = 0.27 dollars per ounce.

The large jar costs less per ounce, so it is the better buy. A unit rate is just a ratio scaled so the bottom equals 1, which makes any two offers directly comparable.

Worked example 7: rearrange the temperature formula

Solve C = (5/9)(F - 32) for F, so you can convert in the other direction.

1. Multiply both sides by the reciprocal 9/5: (9/5)C = F - 32.
2. Add 32 to both sides: F = (9/5)C + 32.

Check with a known pair: water boils at C = 100 and F = 212. Substitute: (9/5)(100) + 32 = 180 + 32 = 212. The rearranged formula reproduces the known fact, so it is correct.

Real-world applications

• Medicine dosing: a drug given at 5 mg per 2 kg of body weight for a 30 kg child: 5/2 = d/30, so cross multiplying gives 2d = 150 and d = 75 mg.
• Currency exchange: if 1 dollar buys 0.90 euros, then 40 dollars buys 40(0.90) = 36 euros; the exchange rate is a constant ratio.
• Cooking: scaling a 4-serving recipe to 10 servings multiplies every ingredient by the ratio 10/4 = 2.5.
• Density, speed, and every "per": any quantity described with the word per (miles per hour, dollars per ounce, students per class) is a ratio, and questions about it are proportions.

Common misconceptions

• Flipping one ratio in a proportion. Miles must stay over gallons on both sides. Writing 150/5 = 8/m instead of 150/5 = m/8 produces a wrong answer; label units top and bottom.
• Cross multiplying across addition. Cross multiplication applies only to one ratio equal to one ratio; it cannot be used on a/b + c/d = e.
• Adding instead of scaling. If 3 eggs make 12 muffins, 20 muffins do not need "3 + 8" eggs; quantities scale by multiplication, so you need 5 eggs because 20/12 = 5/3.
• Believing a variable in the middle of a formula is stuck. Any variable can be isolated with inverse operations, wherever it sits.
• Dropping parentheses when rearranging. In W = (P - 2L)/2, the entire quantity P - 2L is divided by 2, so the parentheses are essential.

Recap

• A ratio compares two quantities; a proportion says two ratios are equal.
• Cross multiplication (ad = bc) solves any proportion, and it is just clearing both denominators.
• Keep the same units in the same positions on both sides of a proportion.
• A formula is an equation relating quantities; solve it for any variable using the usual balancing moves.
• Unit rates (a ratio per 1) make comparisons and conversions immediate.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 2: Solving Linear Equations and Inequalities, section on solving a formula for a specific variable (openstax.org/details/books/elementary-algebra-2e).
• OpenStax, Prealgebra 2e, Chapter 6, section "Solve Proportions and their Applications" (openstax.org/details/books/prealgebra-2e).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Equations With More Than One Variable" (tutorial.math.lamar.edu).

What an inequality means

An inequality compares two expressions that are not necessarily equal. The symbols are < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Instead of one solution, an inequality usually has a whole range of solutions, because many numbers can satisfy a "greater than" or "less than" condition.

You solve inequalities almost exactly like equations, using inverse operations to isolate the variable. There is one crucial extra rule that has no counterpart in equation solving.

The flip rule

When you multiply or divide both sides by a negative number, you must reverse the inequality sign. Here is why: 3 is less than 5, but multiplying both by -1 gives -3 and -5, and now -3 is greater than -5. The order flipped, so the symbol must flip too. Adding or subtracting never triggers the flip; only multiplying or dividing by a negative does.

Worked example 1: a two-step inequality

Solve -2x + 3 > 11.

1. Subtract 3 from both sides: -2x > 8.
2. Divide both sides by -2 and flip the sign: x < -4.

Check with a test value less than -4, say x = -5: -2(-5) + 3 = 13, and 13 > 11 is true. A value not in the solution, x = -3, gives 9, and 9 > 11 is false. The boundary and direction are correct. Testing one value from inside and one from outside the solution is the surest way to confirm you flipped correctly.

Worked example 2: variables on both sides

Solve 3(x + 2) ≤ 4x - 1.

1. Distribute: 3x + 6 ≤ 4x - 1.
2. Subtract 3x from both sides: 6 ≤ x - 1.
3. Add 1 to both sides: 7 ≤ x, which reads x ≥ 7.

Here we divided by nothing negative, so no flip was needed. Reading 7 ≤ x as x ≥ 7 just puts the variable on the left, which is easier to graph.

Graphing on a number line

Show the solution with a ray on the number line. Use an open circle for < or > (the endpoint is not included) and a closed circle for ≤ or ≥ (the endpoint is included). For x < -4, place an open circle at -4 and shade to the left. The shading direction should match where the true test values live, which is another reason to test a point.

Interval notation

In interval notation, x < -4 is written as the interval from negative infinity up to -4 with a parenthesis, and x ≥ 7 is written starting at 7 with a bracket and going to positive infinity. A square bracket includes the endpoint; a parenthesis does not. Infinity always takes a parenthesis, because you can never actually reach it.

Worked example 3: a real constraint

A ride costs 3 dollars to board plus 2 dollars per mile. With 20 dollars, how far can you go? Let m be miles. The cost 3 + 2m must be at most 20: 3 + 2m ≤ 20. Subtract 3: 2m ≤ 17. Divide by 2: m ≤ 8.5 miles. The "at most" wording is what makes it an inequality rather than an equation, and real budgets are full of such limits.

Worked example 4: a flip with a check on both sides

Solve 7 - 3x ≥ 19.

1. Subtract 7 from both sides: -3x ≥ 12.
2. Divide both sides by -3 and flip the inequality: x ≤ -4.

Check inside the solution with x = -5: 7 - 3(-5) = 7 + 15 = 22, and 22 ≥ 19 is true. Check outside with x = 0: 7 - 0 = 7, and 7 ≥ 19 is false. The boundary -4 itself gives 7 + 12 = 19, and 19 ≥ 19 is true, which is why the closed circle includes it. Testing both sides of the boundary confirms the direction of the ray.

Worked example 5: the test-score problem

Your first three test scores are 88, 92, and 85. What must you score on the fourth test to average at least 90? Let x be the fourth score. The average must satisfy:

1. Write the condition: (88 + 92 + 85 + x)/4 ≥ 90.
2. Simplify the sum: (265 + x)/4 ≥ 90.
3. Multiply both sides by 4 (positive, no flip): 265 + x ≥ 360.
4. Subtract 265: x ≥ 95.

You need at least a 95. Check: scores 88 + 92 + 85 + 95 = 360, and 360/4 = 90 exactly. Any score above 95 pushes the average above 90.

Real-world applications

• Budgets: "spend at most 20 dollars" and "keep the balance above zero" are inequalities, not equations.
• Safety limits: an elevator rated for 1000 pounds carrying boxes of 60 pounds each requires 60b ≤ 1000, so b ≤ 16.67, meaning at most 16 whole boxes.
• Eligibility thresholds: minimum heights for rides, passing scores, and speed limits are all boundary values with a direction.
• Business: profit requires revenue greater than cost, an inequality whose solution is the range of sales that keeps a company alive.

Common misconceptions

• Flipping the sign whenever a negative appears. Only multiplying or dividing both sides by a negative flips it; adding or subtracting negatives never does.
• Forgetting to flip when dividing by a negative. From -2x > 8, dividing by -2 gives x < -4, not x > -4.
• Using a closed circle for a strict inequality. x > 2 takes an open circle at 2; the boundary is not included.
• Expecting a single answer. An inequality's solution is a range; -5, -10, and -100 all satisfy x ≤ -4.
• Mixing up bracket and parenthesis in interval notation. A bracket includes its endpoint; a parenthesis excludes it, and infinity always gets a parenthesis.

Recap

• Solve inequalities with the same inverse operations as equations.
• Multiplying or dividing both sides by a negative reverses the inequality sign.
• Graph solutions with open circles for strict inequalities and closed circles when the boundary counts.
• Test one value inside and one outside the solution to confirm the direction.
• Phrases like "at most," "at least," "no more than," and "minimum" signal inequalities in word problems.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 2: Solving Linear Equations and Inequalities, section "Solve Linear Inequalities" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Solving equations and inequalities" unit: multi-step inequalities (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Linear Inequalities" (tutorial.math.lamar.edu).

Two number lines make a plane

The coordinate plane is formed by two number lines that cross at right angles. The horizontal one is the x-axis and the vertical one is the y-axis. They meet at the origin, the point where both values are zero. This simple grid, credited to Rene Descartes, is one of the most important inventions in mathematics because it lets us describe geometry with numbers.

Every point is named by an ordered pair (x, y). The first number tells you how far to move left or right from the origin; the second tells you how far to move up or down. Order matters: (3, 1) is a different point from (1, 3). Reading the pair always goes "across then up", the same way you read a word before you read the line below it.

The four quadrants

The axes split the plane into four regions called quadrants, numbered counterclockwise starting from the top right:

Quadrant	x sign	y sign	Example
I (top right)	+	+	(3, 2)
II (top left)	-	+	(-3, 2)
III (bottom left)	-	-	(-3, -2)
IV (bottom right)	+	-	(3, -2)

Knowing the sign pattern lets you predict a point's quadrant before you plot it, and lets you catch errors: if you meant Quadrant II but your x came out positive, something went wrong.

Worked example: plot a point

To plot (-3, 2), start at the origin, move 3 units left (because x is -3), then move 2 units up (because y is 2). You land in Quadrant II. If either coordinate is 0, the point sits on an axis: (4, 0) is on the x-axis, and (0, -5) is on the y-axis. Points on an axis belong to no quadrant, because they are on the boundary.

Why this matters

The coordinate plane lets us turn equations into pictures. When you list several ordered pairs that satisfy an equation and plot them, a pattern appears, often a straight line, which is the subject of the next lessons. Reading a graph, you can reverse the process: pick a point on the curve and read off its coordinates to recover the numbers it represents. Every scatterplot, map, screen pixel, and video game position is an ordered pair on a coordinate plane.

A quick table of solutions

Consider y = x + 1. An equation in two variables has solutions that are ordered pairs, not single numbers. Choosing a few x-values gives a table:

x	-1	0	1	2
y	0	1	2	3

Plotting (-1, 0), (0, 1), (1, 2), and (2, 3) and connecting them produces a straight line rising to the right. This "make a table, then plot" method works for any equation, and it is the bridge from the algebra you have done to the graphing you are about to learn.

Reading meaning from coordinates

In an applied graph the coordinates carry units. If a graph shows a plant's height over time, the point (3, 12) might mean "at 3 weeks the plant was 12 centimeters tall." Learning to translate a bare pair of numbers into a sentence about the real situation is a skill you will use throughout science.

Worked example 2: build a table and plot a line

Make a table of solutions for y = 2x - 1 and describe the graph.

1. Choose easy x-values: -1, 0, 1, 2.
2. Compute each y: at x = -1, y = 2(-1) - 1 = -3; at x = 0, y = -1; at x = 1, y = 2(1) - 1 = 1; at x = 2, y = 2(2) - 1 = 3.
3. Plot the pairs (-1, -3), (0, -1), (1, 1), (2, 3).

The points fall on a straight line climbing to the right: each step of 1 in x raises y by 2. Verify one pair by substitution: for (2, 3), is 3 = 2(2) - 1? Yes, 4 - 1 = 3, so the point belongs on the graph.

Worked example 3: distances along gridlines

When two points share a coordinate, distance is a subtraction. The points (-2, 3) and (4, 3) share y = 3, so they sit on the same horizontal line; the distance between them is |4 - (-2)| = |6| = 6 units. The points (5, -1) and (5, 6) share x = 5, so the distance is |6 - (-1)| = 7 units. Absolute value appears because distance is always positive regardless of which point you name first.

Real-world applications

• Maps and GPS: latitude and longitude are an ordered pair locating any spot on Earth; a city grid such as "4th Avenue and 12th Street" works the same way.
• Screens and games: every pixel on a display and every character position in a video game is a coordinate pair; game programmers move a sprite by changing its x and y.
• Spreadsheets: cell B7 is column B, row 7, an ordered pair with letters for one axis.
• Science graphs: time on the x-axis and a measured quantity on the y-axis lets one picture summarize hundreds of measurements.

Common misconceptions

• Reversing the coordinates. (3, 1) means right 3 then up 1; going up 3 and right 1 lands on the different point (1, 3).
• Believing every point has a quadrant. Points on an axis, such as (0, 4) or (-6, 0), belong to no quadrant.
• Numbering quadrants clockwise. They go counterclockwise from the top right: I, II, III, IV.
• Reading a graph point as two unrelated numbers. In context, (3, 12) is one fact ("at 3 weeks, 12 cm tall"), not two separate values.
• Assuming a table's pattern without checking. Substitute at least one pair back into the equation to confirm the table row is a true solution.

Recap

• The coordinate plane crosses a horizontal x-axis with a vertical y-axis at the origin (0, 0).
• An ordered pair (x, y) reads "across, then up or down," and order matters.
• Sign patterns identify the four quadrants; axis points belong to none.
• Solutions of a two-variable equation are ordered pairs; a table of them plots into a graph.
• Along a shared gridline, distance is the absolute value of the coordinate difference.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 4: Graphs, section "Use the Rectangular Coordinate System" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, "Coordinate plane" topic pages (khanacademy.org/math).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Graphing" (tutorial.math.lamar.edu).

Steepness as a number

Slope measures how steep a line is and in which direction it tilts. It is the ratio of the vertical change to the horizontal change as you move along the line, often said as rise over run. A large slope means a steep line; a small slope means a gentle one. Slope turns the vague word "steep" into a precise number you can compute and compare.

Given two points (x₁, y₁) and (x₂, y₂), the slope m is:

m = (y₂ - y₁) / (x₂ - x₁)

The rise is the difference in the y-values; the run is the difference in the x-values. Be sure to subtract the coordinates in the same order in both the top and the bottom, using the same point as "first" in each.

Worked example 1: slope from two points

Find the slope of the line through (1, 2) and (4, 11).

1. Rise: 11 - 2 = 9.
2. Run: 4 - 1 = 3.
3. Slope: m = 9/3 = 3.

A slope of 3 means the line rises 3 units for every 1 unit it moves right. You could also read it as 3/1, up 3 and right 1, which is exactly how you would step it out on a graph.

Four kinds of slope

• Positive slope: the line rises from left to right (like the example above).
• Negative slope: the line falls from left to right, such as m = -2.
• Zero slope: a horizontal line; the rise is 0, so m = 0.
• Undefined slope: a vertical line; the run is 0, and dividing by 0 is undefined.

Zero slope and undefined slope are easy to confuse. A flat road has zero slope; a cliff has undefined slope. Remember that a horizontal line still has a defined slope of 0, while a vertical line has no slope value at all.

Worked example 2: a negative slope

Find the slope through (-2, 5) and (2, -3). Rise: -3 - 5 = -8. Run: 2 - (-2) = 4. Slope: m = -8/4 = -2. The line falls as it moves right. Note the run: subtracting a negative became addition, a common place to slip.

Slope as a rate of change

Slope is not only geometric; it is the constant rate of change between two quantities. If a plumber charges by time and the graph of cost versus hours has slope 40, that means the cost rises 40 dollars per hour. If a car's distance-versus-time graph has slope 60, the car travels 60 miles per hour. Whenever one quantity changes at a steady rate with respect to another, the graph is a line and the slope is that rate. The units of slope are always "y-units per x-unit", such as dollars per hour or miles per gallon.

Worked example 3: interpret a real slope

A tank is draining. At 2 minutes it holds 80 liters; at 6 minutes it holds 20 liters. The slope is (20 - 80)/(6 - 2) = -60/4 = -15 liters per minute. The negative sign means the water is decreasing, and the size 15 tells you it drains 15 liters each minute. A real-world slope carries both a direction (the sign) and a rate (the size).

A common mistake to avoid

Do not mix up the order of subtraction. If you compute the rise as y₂ - y₁, the run must be x₂ - x₁, using the same first point on top and bottom. Reversing one but not the other flips the sign and gives the wrong slope. It is fine to start with either point, as long as you are consistent in both the numerator and the denominator.

Worked example 4: slope from a table

A table lists x-values 0, 2, 4, 6 with y-values 5, 8, 11, 14. Is the relationship linear, and if so what is its slope?

1. Slope between the first two rows: (8 - 5)/(2 - 0) = 3/2.
2. Slope between the next rows: (11 - 8)/(4 - 2) = 3/2, and (14 - 11)/(6 - 4) = 3/2.
3. Confirm with the two end rows: (14 - 5)/(6 - 0) = 9/6 = 3/2.

Every pair gives the same slope, 3/2, so the points lie on one line. A constant slope between all pairs of rows is exactly the test for a linear relationship; if even one pair gave a different value, the data would bend.

Real-world applications

• Accessibility ramps: United States accessibility guidelines limit ramps to a slope of 1/12, so a doorway 2 feet above grade needs a run of at least 2 × 12 = 24 feet.
• Road grades: a "6 percent grade" sign means the road rises 6 feet per 100 feet of horizontal distance, a slope of 6/100.
• Wages and rates: on a graph of pay versus hours, slope is the hourly wage; on distance versus time, it is speed; on cost versus items, it is the unit price.
• Draining and filling: negative slopes describe emptying tanks, cooling coffee, and falling account balances; the sign carries the direction of change.

Common misconceptions

• Writing run over rise. Slope is rise over run: the change in y on top, the change in x on the bottom.
• Confusing zero and undefined slope. Horizontal lines have slope 0; vertical lines have undefined slope because the run is 0.
• Mixing subtraction order. If the rise is y2 - y1, the run must be x2 - x1, with the same point first in both.
• Dropping a negative in the run. From x = -2 to x = 2 the run is 2 - (-2) = 4, not 0 or 2.
• Ignoring units. A real-world slope has units, y-units per x-unit, and stating them is part of the answer.

Recap

• Slope m = (y2 - y1)/(x2 - x1), rise over run, measures steepness and direction.
• Positive slopes rise left to right; negative slopes fall; zero is horizontal; undefined is vertical.
• Slope is the constant rate of change, with units of y per x.
• Subtract coordinates in the same order in numerator and denominator.
• Equal slopes between every pair of points is the signature of a linear relationship.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 4: Graphs, section "Understand Slope of a Line" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Linear equations and graphs" unit: slope (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Lines" (tutorial.math.lamar.edu).

Slope-intercept form

The most useful way to write a line is slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept - the y-value where the line crosses the y-axis. This form tells you everything you need to draw the line quickly, because it hands you a starting point (the y-intercept) and a direction (the slope) in one glance.

Worked example 1: graph y = 2x - 3

Here m = 2 and b = -3.

1. Plot the y-intercept at (0, -3).
2. Use the slope 2, which is 2/1: from that point go up 2 and right 1 to reach (1, -1).
3. Repeat to get (2, 1), then draw the straight line through the points.

The line below shows this idea: it crosses the y-axis at -3 and rises steeply.

Using intercepts

For a line written like 2x + 3y = 6, the fastest approach can be the two intercepts.

• x-intercept: set y = 0. Then 2x = 6, so x = 3. The point is (3, 0).
• y-intercept: set x = 0. Then 3y = 6, so y = 2. The point is (0, 2).

Plot both points and draw the line through them. Two intercepts are often the least-arithmetic way to graph an equation in standard form, since setting a variable to zero deletes a term.

Rewriting into slope-intercept form

You can convert 2x + 3y = 6 into y = mx + b. Subtract 2x: 3y = -2x + 6. Divide by 3: y = (-2/3)x + 2. So the slope is -2/3 and the y-intercept is 2, matching the intercept we found. Being able to move between standard form and slope-intercept form lets you pick whichever the problem makes easier.

Horizontal and vertical lines

An equation like y = 4 is a horizontal line through every point with y equal to 4; its slope is 0. An equation like x = -2 is a vertical line through every point with x equal to -2; its slope is undefined. These are easy to spot because only one variable appears. A frequent mix-up is which is which: y = 4 is flat (horizontal), while x = -2 stands straight up (vertical).

Worked example 2: a negative slope and a fractional slope

Graph y = (-1/3)x + 2. Start at the y-intercept (0, 2). The slope -1/3 means down 1 and right 3, landing at (3, 1), then (6, 0). A negative slope tilts the line downward to the right, and a fractional slope makes it gentle. Drawing at least two plotted points before you connect them keeps the line accurate.

Why this matters

A graph shows an entire relationship at a glance: where a business breaks even, how fast something grows, when two plans cost the same. The y-intercept is often a starting amount (a flat fee, an initial population), and the slope is the rate. Reading those two features off an equation, or off a graph, is a skill used constantly in science, business, and everyday budgeting.

Worked example 3: standard form, two ways

Graph 3x - 2y = 6, first by intercepts, then confirm with slope-intercept form.

1. x-intercept: set y = 0: 3x = 6, so x = 2. Point: (2, 0).
2. y-intercept: set x = 0: -2y = 6, so y = -3. Point: (0, -3).
3. Convert: subtract 3x: -2y = -3x + 6. Divide by -2: y = (3/2)x - 3.
4. Confirm: the converted form says start at (0, -3) and go up 3, right 2, which lands exactly on (2, 0). The two methods agree.

Doing a graph two ways like this is the graphing equivalent of checking an equation by substitution: agreement means you can trust the picture.

Worked example 4: read a story from an equation

A water tank holds 50 gallons and drains at 5 gallons per minute: y = -5x + 50, where x is minutes and y is gallons.

1. The y-intercept (0, 50) is the starting amount: 50 gallons at time zero.
2. The slope -5 means the tank loses 5 gallons each minute.
3. The x-intercept comes from 0 = -5x + 50: add 5x to get 5x = 50, so x = 10. The tank is empty at 10 minutes.

Every feature of the graph translates into a sentence about the situation, which is why graphing is a modeling skill and not just a drawing exercise.

Real-world applications

• Startup plus rate: a plumber charging 40 dollars to show up plus 60 per hour is y = 60x + 40; the intercept is the visit fee and the slope is the hourly rate.
• Depreciation: a laptop bought for 900 dollars losing 150 in value each year follows y = -150x + 900, hitting zero value at x = 6 years.
• Science labs: converting between measurement scales and plotting calibration lines both rely on reading slope and intercepts fluently.

Common misconceptions

• Calling x = -2 horizontal. Equations of the form x = constant are vertical lines; y = constant lines are horizontal.
• Reading b as the x-intercept. In y = mx + b, b is the y-intercept; find the x-intercept by setting y = 0.
• Plotting the slope from the origin. The slope steps start at the y-intercept, not at (0, 0), unless the intercept happens to be zero.
• Inverting a fractional slope. A slope of -1/3 means down 1 and right 3, not down 3 and right 1.
• Connecting only one plotted point. A reliable line needs at least two points, and a third is a cheap error check.

Recap

• Slope-intercept form y = mx + b hands you the starting point (0, b) and the stepping direction m.
• For standard form Ax + By = C, the intercepts (set y = 0, then x = 0) are usually the fastest route.
• Convert between forms by solving for y; both describe the same line.
• y = constant is horizontal, x = constant is vertical.
• In applications, the y-intercept is the starting value and the slope is the rate of change.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 4: Graphs, sections "Graph Linear Equations in Two Variables," "Graph with Intercepts," and "Use the Slope-Intercept Form" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Linear equations and graphs" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Lines" (tutorial.math.lamar.edu).

Three situations, one goal

Writing the equation of a line means finding y = mx + b from the information you are given. There are three common situations, and they all reduce to finding the slope m and the intercept b. Once you have those two numbers, the equation writes itself.

Case 1: given slope and y-intercept

This is the easiest: just plug the numbers into y = mx + b. If the slope is 4 and the y-intercept is -1, the equation is y = 4x - 1. No work is needed beyond substitution.

Case 2: given a point and a slope (point-slope form)

The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is the known point and m is the slope. Substitute the values, then simplify into slope-intercept form. Point-slope exists precisely because you often know a point that is not the y-intercept.

Worked example: write the line with slope 3 through the point (1, 2).

1. Point-slope: y - 2 = 3(x - 1).
2. Distribute: y - 2 = 3x - 3.
3. Add 2 to both sides: y = 3x - 1.

Check: does the point (1, 2) satisfy it? 3(1) - 1 = 2. Yes.

Case 3: given two points

When you have two points, first compute the slope, then use point-slope form with either point. The slope is what unlocks the rest.

Worked example: write the line through (2, 1) and (4, 7).

1. Slope: m = (7 - 1)/(4 - 2) = 6/2 = 3.
2. Point-slope with (2, 1): y - 1 = 3(x - 2).
3. Distribute and simplify: y - 1 = 3x - 6, so y = 3x - 5.

Check the other point (4, 7): 3(4) - 5 = 7. Correct, so both points lie on y = 3x - 5. Using the other point in step 2 would give the same final equation, which is a good way to verify your work.

Parallel and perpendicular lines

Two useful facts help you write related lines:

• Parallel lines have the same slope. A line parallel to y = 2x + 5 also has slope 2, for example y = 2x - 3.
• Perpendicular lines have slopes that are negative reciprocals - flip the fraction and change the sign, so their product is -1. A line perpendicular to y = 2x + 5 has slope -1/2.

Worked example: a parallel line through a point

Write the line parallel to y = 2x + 5 that passes through (3, 4). Parallel means the slope is still 2. Use point-slope: y - 4 = 2(x - 3). Distribute: y - 4 = 2x - 6. Add 4: y = 2x - 2. It has the same slope as the original but a different y-intercept, exactly as parallel lines should.

Why this matters

Writing an equation from data is how you build a model. Given two data points, such as a startup cost and a later total, you can find the line that predicts every value in between and beyond. Perpendicular lines matter in geometry and design, from the corners of a building to the shortest path from a point to a line. With these tools you can turn almost any description of a linear relationship into an equation you can compute with.

Worked example: a perpendicular line through a point

Write the equation of the line perpendicular to y = (1/3)x - 2 that passes through (2, 1).

1. The given slope is 1/3, so the perpendicular slope is the negative reciprocal: flip to 3/1 and change the sign, giving m = -3.
2. Point-slope form with (2, 1): y - 1 = -3(x - 2).
3. Distribute: y - 1 = -3x + 6.
4. Add 1 to both sides: y = -3x + 7.

Check: the point: -3(2) + 7 = -6 + 7 = 1, correct. The slopes: (1/3)(-3) = -1, so the lines really are perpendicular.

Worked example: a linear model from two data points

A gym membership cost 95 dollars in total after 2 months and 200 dollars in total after 5 months. Model the total cost y after x months.

1. Slope from the data points (2, 95) and (5, 200): m = (200 - 95)/(5 - 2) = 105/3 = 35 dollars per month.
2. Find b using (2, 95): 95 = 35(2) + b, so 95 = 70 + b and b = 25.
3. The model is y = 35x + 25.

Check the other point: 35(5) + 25 = 175 + 25 = 200. Correct. The model also tells a story: 25 dollars was the joining fee and 35 dollars is the monthly rate, numbers that were never stated directly but fall out of the algebra.

Real-world applications

• Pricing and fees: two receipts are enough to recover a company's flat fee and per-unit rate, exactly as in the gym example.
• Depreciation: a car worth 24,000 dollars new and losing 3,000 per year follows y = -3000x + 24000.
• Science: two calibrated readings determine the conversion line between instrument units and true values.
• Construction and design: perpendicular slopes verify square corners on plans drawn in coordinates.

Common misconceptions

• Using the opposite instead of the negative reciprocal. Perpendicular to slope 2 is -1/2, not -2; the product of perpendicular slopes must be -1.
• Sign slips inside point-slope form. Through (2, -5) with slope 3, the form is y - (-5) = 3(x - 2), which becomes y + 5 on the left.
• Thinking the two given points lead to different lines. Either point in point-slope form simplifies to the same equation.
• Forgetting parallel lines share slope but not intercept. A parallel line through a new point keeps m and gets its own b.
• Stopping in point-slope form when slope-intercept is requested. Distribute and isolate y to finish.

Recap

• Slope plus y-intercept: plug directly into y = mx + b.
• Point plus slope: use y - y1 = m(x - x1), then simplify.
• Two points: compute the slope first, then apply point-slope with either point.
• Parallel lines share slopes; perpendicular slopes are negative reciprocals.
• Check every equation by substituting the given points back in.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 4: Graphs, section "Find the Equation of a Line" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Forms of linear equations" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Lines" (tutorial.math.lamar.edu).

What is a system?

A system of equations is two or more equations considered together. For two lines, the solution is the ordered pair (x, y) that satisfies both equations at once - graphically, the point where the two lines cross. A single linear equation in two variables has infinitely many solutions (a whole line of them); pairing it with a second equation usually narrows that down to one point.

There are three possibilities. If the lines cross once, there is exactly one solution. If they are parallel and never meet, there is no solution. If they are the same line, there are infinitely many solutions. These match the three outcomes you saw for single equations, which is no coincidence.

Method 1: graphing

Graph both lines on the same axes and read off the intersection point. Take y = 2x + 1 and y = -x + 7. Graphing both, they cross at (2, 5). Graphing is intuitive and shows the whole picture, but it is imprecise when the answer is not a whole number, so we also use algebra to get exact answers.

Method 2: substitution

Substitution works best when one equation is already solved for a variable. You substitute that expression into the other equation, leaving a single-variable equation to solve. The strategy trades a two-variable problem for a one-variable problem you already know how to handle.

Worked example 1: substitution

Solve the system y = 2x + 1 and y = -x + 7.

1. Both equal y, so set them equal: 2x + 1 = -x + 7.
2. Add x to both sides: 3x + 1 = 7.
3. Subtract 1: 3x = 6, so x = 2.
4. Substitute x = 2 into either equation: y = 2(2) + 1 = 5.

The solution is (2, 5). Check in the second equation: -2 + 7 = 5. Both equations give y = 5, so it is correct. Always find both coordinates; a system's answer is a point, not a single number.

Worked example 2: substitute an expression

Solve x + y = 10 and y = 3x - 2. The second equation is already solved for y, so substitute it into the first: x + (3x - 2) = 10. Combine: 4x - 2 = 10, so 4x = 12 and x = 3. Then y = 3(3) - 2 = 7. The solution is (3, 7). Check: 3 + 7 = 10. Correct.

Worked example 3: a word problem

Two numbers add to 20, and the larger is 4 more than the smaller. Let the smaller be x and the larger be y. Then x + y = 20 and y = x + 4. Substitute: x + (x + 4) = 20, so 2x + 4 = 20, giving x = 8 and y = 12. The numbers are 8 and 12. Translating a story into two equations and solving by substitution is one of the most common real uses of systems.

When to pick substitution

Reach for substitution whenever one equation has a variable already isolated, or when isolating a variable is easy (its coefficient is 1). If neither variable is easy to isolate, the elimination method in the next lesson is usually faster. Choosing the right method is itself a skill; a good solver looks at the system before diving in.

Worked example 4: isolate first, then substitute

Solve the system 2x + y = 11 and 3x - 2y = 6. Neither equation is solved for a variable yet, but y in the first has coefficient 1, so isolate it there.

1. From the first equation: y = 11 - 2x.
2. Substitute into the second: 3x - 2(11 - 2x) = 6.
3. Distribute the -2: 3x - 22 + 4x = 6.
4. Combine: 7x - 22 = 6, so 7x = 28 and x = 4.
5. Back-substitute: y = 11 - 2(4) = 11 - 8 = 3.

The solution is (4, 3). Check in both equations: 2(4) + 3 = 11 and 3(4) - 2(3) = 12 - 6 = 6. Both hold, so the point truly lies on both lines. Note the parentheses around 11 - 2x in step 2; substituting without them is the classic way to lose the -2 on the second term.

Worked example 5: a ticket-sales model

A theater sells adult tickets for 10 dollars and child tickets for 6 dollars. One night it sells 50 tickets for 420 dollars. How many of each? Let a be adult and c be child tickets.

1. Count equation: a + c = 50. Money equation: 10a + 6c = 420.
2. Isolate a: a = 50 - c.
3. Substitute: 10(50 - c) + 6c = 420, so 500 - 10c + 6c = 420.
4. Combine: 500 - 4c = 420, so -4c = -80 and c = 20.
5. Then a = 50 - 20 = 30.

Check: 30 + 20 = 50 tickets, and 10(30) + 6(20) = 300 + 120 = 420 dollars. The night sold 30 adult and 20 child tickets.

Real-world applications

• Break-even analysis: where a cost line meets a revenue line is a system's solution; it tells a business how many units make the venture profitable.
• Mixtures and totals: two unknowns with two facts (a count and a value) appear in ticket sales, coin problems, and blending prices.
• Navigation and games: finding where two straight paths cross is literally solving a system of two line equations.

Common misconceptions

• Reporting only x. The solution is the ordered pair (x, y); finish with back-substitution.
• Substituting without parentheses. Replacing y with 11 - 2x inside -2y must produce -2(11 - 2x); skipping the parentheses drops a term.
• Substituting into the same equation you isolated from. That returns a useless identity like 11 = 11; always substitute into the other equation.
• Trusting a graph for exact answers. Non-integer intersections need algebra; graphs give estimates.
• Checking in only one equation. A solution must satisfy both; a point that fits one line only is not a solution of the system.

Recap

• A system's solution is the ordered pair satisfying every equation, the crossing point of the lines.
• One crossing means one solution; parallel lines mean none; identical lines mean infinitely many.
• Substitution: isolate a variable (easiest when its coefficient is 1), substitute into the other equation, solve, back-substitute.
• Translate word problems into a count equation and a value equation.
• Check the pair in both original equations.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 5: Systems of Linear Equations, sections "Solve Systems of Equations by Graphing" and "Solve Systems of Equations by Substitution" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Systems of equations" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Linear Systems with Two Variables" (tutorial.math.lamar.edu).

The idea behind elimination

The elimination method (also called the addition method) combines the two equations so that one variable cancels out. If the coefficients of one variable are opposites, adding the equations makes that variable disappear, leaving one equation in one variable. The whole aim is to engineer a pair of opposite coefficients so a variable vanishes on contact.

Worked example 1: scale, then add

Solve 2x + 3y = 12 and x - y = 1. The coefficients do not cancel yet. Multiply the second equation by 3 so the y-terms become opposites: 3x - 3y = 3. Now the y-terms are +3y and -3y.

1. Add the two equations: (2x + 3y) + (3x - 3y) = 12 + 3.
2. The y-terms cancel: 5x = 15, so x = 3.
3. Substitute into x - y = 1: 3 - y = 1, so y = 2.

The solution is (3, 2). Check in the first equation: 2(3) + 3(2) = 6 + 6 = 12. Correct.

Worked example 2: subtracting equations

Solve 3x + 2y = 16 and 3x - y = 4. The x-coefficients are already equal, so subtract the second equation from the first to eliminate x:

1. (3x + 2y) - (3x - y) = 16 - 4.
2. The x-terms cancel: 3y = 12, so y = 4.
3. Substitute into 3x - y = 4: 3x - 4 = 4, so 3x = 8 and x = 8/3.

The solution is (8/3, 4). Elimination handles fraction answers cleanly, which is one of its advantages over graphing. When subtracting, distribute the minus sign to every term of the second equation, a common place for sign errors.

Multiplying both equations

Sometimes you must scale both equations. For 2x + 3y = 7 and 3x + 2y = 8, multiply the first by 3 and the second by 2 to make the x-coefficients 6 and 6: 6x + 9y = 21 and 6x + 4y = 16. Subtracting gives 5y = 5, so y = 1; then 2x + 3 = 7, so x = 2. The solution is (2, 1). Choosing what to multiply by is like finding a common denominator for the coefficients.

Worked example 3: a mixture problem

A concession stand sells small drinks for 2 dollars and large drinks for 3 dollars. One evening it sells 100 drinks for 240 dollars. How many of each? Let s be small and l be large. Then s + l = 100 and 2s + 3l = 240. Multiply the first by 2: 2s + 2l = 200. Subtract from the second: l = 40. Then s = 60. So 60 small and 40 large. Two-unknown word problems like this are the natural home of elimination.

Choosing a method

Use substitution when a variable is already isolated or has a coefficient of 1. Use elimination when the equations are lined up in standard form and one variable can be canceled with simple scaling. Both always give the same answer; pick whichever needs less work. Spending a moment to choose the easier method often saves more time than it costs.

Worked example 4: scale with a negative multiplier

Solve 4x + 3y = 10 and 2x + y = 4. Multiplying the second equation by -3 turns its y-term into the opposite of +3y:

1. Scale the second equation by -3: -6x - 3y = -12.
2. Add it to the first: (4x + 3y) + (-6x - 3y) = 10 + (-12).
3. The y-terms cancel: -2x = -2, so x = 1.
4. Back-substitute into 2x + y = 4: 2(1) + y = 4, so y = 2.

The solution is (1, 2). Check in the first equation: 4(1) + 3(2) = 4 + 6 = 10. Correct. A negative multiplier lets you always add the equations, which many students find less error-prone than subtracting.

Worked example 5: menu prices from two receipts

Two burgers and three orders of fries cost 16 dollars; two burgers and one order of fries cost 12 dollars. Find each price. Let b be a burger and f be fries.

1. The system: 2b + 3f = 16 and 2b + f = 12.
2. The b-coefficients already match, so subtract the second equation from the first: (2b + 3f) - (2b + f) = 16 - 12.
3. The b-terms cancel: 2f = 4, so f = 2.
4. Back-substitute: 2b + 2 = 12, so 2b = 10 and b = 5.

Check the first receipt: 2(5) + 3(2) = 10 + 6 = 16. Burgers cost 5 dollars and fries cost 2 dollars. Real prices recovered from two receipts, with no menu in sight.

Real-world applications

• Recovering hidden rates: two mixed purchases determine two unknown unit prices, as in the receipts example.
• Mixture chemistry: blending two solutions to hit a target concentration produces a system solved by elimination.
• Large-scale computing: the same idea, systematically eliminating one variable at a time, is Gaussian elimination, the algorithm computers use on systems with thousands of unknowns in engineering and weather models.

Common misconceptions

• Scaling only one side. Multiplying an equation by 3 multiplies every term on both sides, including the constant.
• Subtracting carelessly. The minus distributes to every term: (3x + 2y) - (3x - y) gives 3y, because subtracting -y adds y.
• Adding equations when nothing cancels. Adding is only useful once one variable has opposite coefficients; scale first if needed.
• Forgetting the second coordinate. After finding x, substitute back to get y; the answer is a pair.
• Thinking elimination and substitution can disagree. They are different routes to the same unique answer; a disagreement signals an arithmetic error.

Recap

• Elimination adds or subtracts equations so one variable cancels.
• Scale one or both equations to create opposite coefficients; a negative multiplier lets you add instead of subtract.
• Solve the surviving one-variable equation, then back-substitute.
• Prefer substitution when a variable is isolated; prefer elimination for lined-up standard forms.
• Check the final pair in both original equations.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 5: Systems of Linear Equations, sections "Solve Systems of Equations by Elimination" and "Solve Applications with Systems of Equations" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Systems of equations" unit: elimination method (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Linear Systems with Two Variables" (tutorial.math.lamar.edu).

What an exponent means

An exponent tells you how many times to multiply a base by itself. In x⁵, the base is x and the exponent is 5, so x⁵ = x · x · x · x · x. Understanding this meaning lets every rule below make sense rather than being memorized blindly. Exponents are a compact way to write repeated multiplication, just as multiplication is a compact way to write repeated addition.

The three main rules

• Product rule: when multiplying like bases, add the exponents: xᵀ · xⁿ = xᵀ⁺ⁿ. Example: x³ · x⁴ = x⁷. It works because you are just counting all the factors of x: three of them times four of them is seven of them.
• Quotient rule: when dividing like bases, subtract the exponents: xᵀ ÷ xⁿ = xᵀ⁻ⁿ. Example: x⁷ ÷ x² = x⁵. Two factors on the bottom cancel two on the top, leaving five.
• Power rule: when raising a power to a power, multiply the exponents: (xᵀ)ⁿ = xᵀⁿ. Example: (x⁴)³ = x¹². Three groups of four factors is twelve factors.

Notice all three rules come straight from the meaning of an exponent as repeated multiplication. If you ever forget a rule, write out the factors and the rule reappears.

Coefficients and multiple variables

When terms have number coefficients, handle the numbers separately from the variables. Example: (2x³)(4x²). Multiply the coefficients: 2 × 4 = 8. Add the exponents of x: 3 + 2 = 5. Result: 8x⁵. Coefficients are multiplied, never added, while exponents of like bases are added; keeping those two operations straight prevents most errors.

With a product raised to a power, the exponent reaches every factor: (3x²y)² = 3² · x⁴ · y² = 9x⁴y². Notice the coefficient 3 is also squared. Forgetting to square the coefficient is a classic slip.

Zero and negative exponents

• Zero exponent: any nonzero base to the power 0 equals 1: x⁰ = 1. This follows from the quotient rule, since xᵀ ÷ xᵀ = x⁰ and any nonzero number divided by itself is 1.
• Negative exponent: a negative exponent means a reciprocal: x⁻ⁿ = 1 / xⁿ. Example: x⁻³ = 1/x³, and 2⁻² = 1/2² = 1/4. A negative exponent does not make the number negative; it moves the factor to the denominator.

Worked example: combine several rules

Simplify (x⁴)³ · x⁻² ÷ x⁵.

1. Power rule: (x⁴)³ = x¹².
2. Product rule with x⁻²: x¹² · x⁻² = x¹⁰ (12 + (-2) = 10).
3. Quotient rule dividing by x⁵: x¹⁰ ÷ x⁵ = x⁵ (10 - 5 = 5).

The simplified result is x⁵. Work one rule at a time and the exponents stay under control.

Why this matters

Exponents are the language of growth and scale. Compound interest, population growth, radioactive decay, the loudness of sound, and the magnitude of earthquakes are all described with exponents. The laws you are learning also power scientific notation, which lets scientists write the distance to a star or the size of an atom without a page full of zeros. Every one of these applications relies on the same product, quotient, and power rules.

Worked example: quotients with coefficients

Simplify 12x8 / (3x5).

1. Divide the coefficients: 12 / 3 = 4.
2. Apply the quotient rule to the variable: x8 / x5 = x8-5 = x3.
3. Result: 4x³.

Check by expansion: eight factors of x on top and five on the bottom cancel to leave three on top, and 12 divided by 3 is 4. The shortcut and the long way agree.

Worked example: a power of a product

Simplify (2x3y)4. The exponent 4 applies to every factor inside:

1. Coefficient: 24 = 16.
2. The x-part by the power rule: (x3)4 = x12.
3. The y-part: y4.

Result: 16x¹²y⁴. Verify the coefficient: 2 × 2 = 4, × 2 = 8, × 2 = 16. Forgetting to raise the 2 is the single most common slip on problems like this.

Real-world applications

• Doubling and growth: bacteria that double every hour multiply by 26 = 64 in six hours; the same doubling law governs compound interest and viral sharing.
• Computer science: memory sizes run on powers of 2: 210 = 1024, which is why a kilobyte is not exactly a thousand bytes.
• Scientific notation: the speed of light is about 3 × 108 meters per second; multiplying and dividing such numbers uses the product and quotient rules on the powers of 10.
• Scaling laws: doubling the side of a square multiplies its area by 22 = 4; doubling a cube's side multiplies volume by 23 = 8.

Common misconceptions

• Multiplying exponents in a product. x2 · x3 = x5 (add), not x⁶; count the factors to see why.
• Adding exponents in a power of a power. (x4)3 = x12 (multiply), not x⁷.
• Reading 2³ as 2 × 3. It is 2 · 2 · 2 = 8, not 6.
• Believing x⁰ = 0. Any nonzero base to the zero power is 1, forced by the quotient rule.
• Squaring only the variable in (3x)². The exponent hits every factor: (3x)2 = 9x2, not 3x².

Recap

• Product rule: add exponents of like bases. Quotient rule: subtract. Power rule: multiply.
• Handle coefficients with ordinary arithmetic, separately from the exponents.
• A zero exponent gives 1; a negative exponent moves the factor across the fraction bar.
• An outer exponent applies to every factor inside the parentheses, including the coefficient.
• When in doubt, expand into repeated factors; every rule reappears from the definition.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 6: Polynomials, sections "Use Multiplication Properties of Exponents," "Divide Monomials," and "Integer Exponents and Scientific Notation" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Exponents and radicals" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Integer Exponents" (tutorial.math.lamar.edu).

What a polynomial is

A polynomial is a sum of terms, where each term is a number times a variable raised to a whole-number power. Examples: 3x² + 5x - 2 and 7x - 4. Special names by number of terms: a monomial has one term, a binomial has two, and a trinomial has three. Expressions with variables in a denominator or under a root are not polynomials, because their exponents are not whole numbers.

The degree of a polynomial is the highest exponent that appears. In 3x² + 5x - 2, the degree is 2. A degree-2 polynomial is called quadratic, which leads into the final module; degree 1 is linear, and degree 0 is a constant. Writing a polynomial in order from highest to lowest degree, called standard form, makes it easier to read and to add.

Adding and subtracting

To add or subtract polynomials, combine like terms. When subtracting, distribute the negative sign to every term in the second polynomial first. This single habit prevents the most common polynomial mistake.

Worked example 1: subtraction

Compute (4x² + 3x - 5) - (x² - 2x + 1).

1. Distribute the minus sign: 4x² + 3x - 5 - x² + 2x - 1.
2. Combine x² terms: 4x² - x² = 3x².
3. Combine x terms: 3x + 2x = 5x.
4. Combine constants: -5 - 1 = -6.

Result: 3x² + 5x - 6. Notice how -(-2x) became +2x; missing that sign flip is where errors creep in.

Multiplying a monomial by a polynomial

Distribute the monomial to each term. Example: 2x(3x² - 4x + 1) = 6x³ - 8x² + 2x. Each term picks up an extra factor of x, so its exponent rises by 1, and each coefficient is multiplied by 2. Every term inside must receive the outside factor.

Multiplying two binomials with FOIL

FOIL is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then combine. FOIL is really just the distributive property applied twice, organized so you do not miss a product.

Worked example 2: FOIL

Compute (x + 3)(x - 5).

• First: x · x = x².
• Outer: x · (-5) = -5x.
• Inner: 3 · x = 3x.
• Last: 3 · (-5) = -15.

Combine the middle terms: -5x + 3x = -2x. Result: x² - 2x - 15. The two middle terms almost always combine, which is why the product of two binomials usually has three terms.

Worked example 3: a perfect square

Compute (x - 4)² = (x - 4)(x - 4). FOIL gives x² - 4x - 4x + 16 = x² - 8x + 16. Squaring a binomial always produces a trinomial like this, following the pattern (a - b)² = a² - 2ab + b². Recognizing that pattern will speed up your factoring later.

Why this matters

Polynomials model areas, volumes, projectile paths, profit curves, and countless engineering relationships. Multiplying binomials is the exact reverse of the factoring you will do next, and factoring is the key to solving quadratic equations. So the FOIL skill you build here pays off immediately, both forward (expanding) and backward (factoring).

Worked example 4: binomial times trinomial

Multiply (x + 2)(x2 + 3x + 4). FOIL does not apply (the second factor has three terms), so distribute each term of the binomial across the trinomial:

1. Distribute the x: x · x2 + x · 3x + x · 4 = x3 + 3x2 + 4x.
2. Distribute the 2: 2 · x2 + 2 · 3x + 2 · 4 = 2x2 + 6x + 8.
3. Add the two rows and combine like terms: x3 + (3x2 + 2x2) + (4x + 6x) + 8.

Result: x³ + 5x² + 10x + 8. Every one of the 2 × 3 = 6 partial products appeared exactly once; organized distribution scales to any sizes.

Worked example 5: an area model you can verify

A rectangular garden is (x + 5) meters long and (x + 3) meters wide, so its area is (x + 5)(x + 3). FOIL: x2 + 3x + 5x + 15 = x2 + 8x + 15.

Verify with a number: let x = 10. The dimensions become 15 by 13 meters, and 15 × 13 = 195 square meters. The formula gives 102 + 8(10) + 15 = 100 + 80 + 15 = 195. The polynomial and the direct measurement agree, which is exactly what an algebraic identity promises: it is true for every value of x.

Real-world applications

• Areas and volumes: expanding products of dimensions gives area and volume formulas when sizes depend on an unknown.
• Projectile motion: heights of thrown objects follow degree-2 polynomials in time, such as h = -16t2 + 48t + 4 in feet and seconds.
• Business models: revenue is price times quantity; when a price change affects sales, multiplying the two binomials produces a quadratic revenue polynomial.
• Computing: multiplying multi-digit numbers digit by digit mirrors polynomial multiplication term by term.

Common misconceptions

• Squaring term by term. (x - 4)2 is x² - 8x + 16, not x² + 16 or x² - 16; the middle term -8x comes from the two cross products and cannot be skipped.
• Negating only the first term when subtracting. The minus sign flips every term of the subtracted polynomial.
• Combining unlike powers. x² + x³ stays as is; different exponents mean unlike terms.
• Counting degree by the number of terms. Degree is the highest exponent, not how many terms appear.
• Forcing FOIL onto trinomials. FOIL only lists the four products of two binomials; bigger factors need full distribution.

Recap

• Polynomials are sums of whole-number-power terms; degree is the highest exponent.
• Add and subtract by combining like terms, distributing any minus sign completely.
• Multiply by distributing every term of one factor across every term of the other; FOIL is the two-term shortcut.
• Perfect squares follow (a - b)² = a² - 2ab + b².
• Check an expansion by substituting a small number into both forms.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 6: Polynomials, sections "Add and Subtract Polynomials," "Multiply Polynomials," and "Special Products" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Quadratics: Multiplying and factoring" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Polynomials" (tutorial.math.lamar.edu).

Factoring reverses multiplication

Factoring means writing a polynomial as a product of simpler polynomials - it is the reverse of the multiplying you just learned. If multiplying is like combining ingredients into a dish, factoring is like naming the ingredients from a taste. Factoring is the key to solving quadratic equations, coming next, because it rewrites a sum as a product that a special property can crack open.

Step 1: greatest common factor

Always look first for a greatest common factor (GCF) shared by every term, and pull it out front. Example: 6x³ + 9x². The GCF is 3x², so 6x³ + 9x² = 3x²(2x + 3). You can check by distributing back, which should return the original. Pulling out the GCF first often makes what remains much easier to factor.

Factoring x² + bx + c

To factor a trinomial like x² + bx + c, find two numbers that multiply to c and add to b. Then the factors are (x + first)(x + second). This works because when you FOIL (x + p)(x + q) you get x² + (p + q)x + pq, so the middle coefficient is the sum and the constant is the product.

Worked example 1: positive constant

Factor x² + 7x + 12. Look for two numbers that multiply to 12 and add to 7. Test pairs: 3 and 4 multiply to 12 and add to 7. So x² + 7x + 12 = (x + 3)(x + 4). Check with FOIL: x² + 4x + 3x + 12 = x² + 7x + 12. Correct.

Worked example 2: a negative constant

Factor x² - 5x - 14. We need two numbers that multiply to -14 and add to -5. Since the product is negative, the numbers have opposite signs. Test -7 and 2: they multiply to -14 and add to -5. So the factors are (x - 7)(x + 2). Reading the signs of b and c tells you the signs of the two numbers before you even start guessing.

Difference of two squares

A difference of squares has the special pattern a² - b² = (a - b)(a + b). It appears whenever you subtract one perfect square from another. There is no middle term because the two cross products cancel.

Worked example 3: difference of squares

Factor x² - 9. Since 9 is 3², this is x² - 3² = (x - 3)(x + 3). Similarly, 4x² - 25 = (2x)² - 5² = (2x - 5)(2x + 5). A sum of squares like x² + 9 does not factor with real numbers, so watch the sign.

Combining techniques

Often you factor out a GCF first, then factor what remains. Example: 2x² + 10x + 12. Pull out 2: 2(x² + 5x + 6). Now factor the trinomial: numbers that multiply to 6 and add to 5 are 2 and 3, so the full factorization is 2(x + 2)(x + 3). Always factor completely, checking whether any remaining factor can be factored further.

Why this matters

Factoring is the reverse gear of algebra, and it is essential for solving equations, simplifying fractions with polynomials, and analyzing graphs. The x-values where a factored expression equals zero are exactly the points where its graph crosses the x-axis, which you will use throughout the quadratics module and in every later math course that touches functions.

Worked example 4: GCF first, then difference of squares

Factor 5x3 - 45x completely.

1. Find the GCF of both terms: 5 divides both coefficients and x divides both terms, so the GCF is 5x.
2. Pull it out: 5x(x2 - 9).
3. The leftover x2 - 9 is a difference of squares: (x - 3)(x + 3).
4. Full factorization: 5x(x - 3)(x + 3).

Check by expanding: (x - 3)(x + 3) = x2 - 9, and 5x(x2 - 9) = 5x3 - 45x, the original. Stopping at step 2 would not be factored completely.

Worked example 5: a first taste of the ac method

Factor 2x2 + 7x + 3, where the leading coefficient is not 1.

1. Multiply a times c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to the middle coefficient 7: they are 6 and 1.
2. Split the middle term using them: 2x2 + 6x + x + 3.
3. Factor by grouping: from the first pair take out 2x: 2x(x + 3); from the second pair take out 1: 1(x + 3).
4. Both groups share (x + 3): the factorization is (2x + 1)(x + 3).

Check with FOIL: 2x2 + 6x + x + 3 = 2x2 + 7x + 3. Correct. The same "multiply to this, add to that" thinking still runs the show; only the bookkeeping grew.

Real-world applications

• Landing times: a ball thrown upward with height h = -16t2 + 64t factors as -16t(t - 4), so h = 0 at t = 0 (launch) and t = 4 seconds (landing).
• Dimensions from area: a floor plan with area x2 + 7x + 12 square feet factors into sides (x + 3) and (x + 4).
• Simplifying formulas: engineers factor shared quantities out of long expressions to make formulas shorter, cheaper to compute, and easier to read, the same GCF move you practiced first.

Common misconceptions

• Skipping the GCF. Always remove common factors first; the remaining trinomial is smaller and friendlier.
• Ignoring the signs of b and c. A positive c means both numbers share a sign (the sign of b); a negative c means opposite signs. Reading this first shrinks the search.
• Factoring a sum of squares. x2 + 9 does not factor over the real numbers; only differences of squares split.
• Stopping too early. After factoring, inspect each factor; expressions like 2(x2 - 4) still hide a difference of squares.
• Never checking. FOIL the answer; if it does not rebuild the original polynomial, a sign or pair was wrong.

Recap

• Factoring rewrites a sum as a product, reversing multiplication.
• Order of attack: GCF first, then trinomial patterns, then difference of squares, and repeat until nothing factors further.
• For x² + bx + c, find two numbers that multiply to c and add to b.
• a² - b² = (a - b)(a + b); sums of squares stay whole.
• Verify every factorization by expanding it back out.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 7: Factoring, sections on the greatest common factor, factoring trinomials, and special products (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Quadratics: Multiplying and factoring" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Factoring Polynomials" (tutorial.math.lamar.edu).

What a quadratic equation is

A quadratic equation is one that can be written in the standard form ax² + bx + c = 0, where a is not zero. The presence of an x² term (and no higher power) makes it quadratic. Unlike a linear equation, a quadratic often has two solutions, because a parabola can cross the x-axis in two places.

The zero-product property

The heart of factoring to solve is the zero-product property: if a product of factors equals zero, then at least one of the factors must be zero. In symbols, if A · B = 0, then A = 0 or B = 0. This turns a factored quadratic into two simple linear equations. The property is special to zero: knowing a product equals 12 tells you nothing about the individual factors, but a product of zero forces one of them to be zero.

Worked example 1: solve by factoring

Solve x² - 5x + 6 = 0.

1. Factor the left side: find numbers that multiply to 6 and add to -5, which are -2 and -3. So (x - 2)(x - 3) = 0.
2. Apply the zero-product property: x - 2 = 0 or x - 3 = 0.
3. Solve each: x = 2 or x = 3.

Check both: for x = 2, 4 - 10 + 6 = 0. For x = 3, 9 - 15 + 6 = 0. Both work, so the solutions are 2 and 3.

Worked example 2: rearrange first

Solve x² + 3x = 4. The equation must equal zero before you factor, so subtract 4 from both sides: x² + 3x - 4 = 0. Factor: numbers that multiply to -4 and add to 3 are 4 and -1, giving (x + 4)(x - 1) = 0. So x = -4 or x = 1. Never apply the zero-product property until one side is truly zero; factoring a nonzero side and setting factors equal to the other number is a serious error.

Worked example 3: a common factor

Solve x² - 7x = 0. Factor out the GCF x: x(x - 7) = 0. Then x = 0 or x - 7 = 0, giving x = 0 or x = 7. Do not divide both sides by x, or you would lose the solution x = 0. Dividing away a variable is one of the most common ways students accidentally throw out a real answer.

Difference of squares in equations

Solve x² - 9 = 0. Factor as a difference of squares: (x - 3)(x + 3) = 0, so x = 3 or x = -3. These two solutions are opposites, which always happens for x² = a positive number.

The reliable routine and why it matters

The two-step habit - get zero on one side, factor, then set each factor to zero - solves every quadratic you can factor. Quadratics describe the path of a thrown ball, the area of a rectangle with a fixed perimeter, and the point where revenue peaks. Setting such a model equal to zero and factoring finds exactly when the ball lands, when the area hits a target, or when profit is zero, so this routine is a genuine problem-solving tool, not just an exercise.

Worked example 4: when does the ball land?

A ball is thrown upward so its height in feet after t seconds is h = -16t2 + 48t. When does it return to the ground?

1. Landing means height zero: -16t2 + 48t = 0.
2. Factor out the GCF, -16t: -16t(t - 3) = 0.
3. Zero-product property: -16t = 0 or t - 3 = 0, so t = 0 or t = 3.

The solution t = 0 is the launch moment; the meaningful answer is t = 3 seconds. Check: -16(3)2 + 48(3) = -16(9) + 144 = -144 + 144 = 0. Interpreting which root answers the question is part of the solution in applied problems.

Worked example 5: dimensions from an area

A rectangle's length is 3 meters more than its width, and its area is 40 square meters. Find the dimensions. Let w be the width, so the length is w + 3.

1. Area equation: w(w + 3) = 40.
2. Expand and set to zero: w2 + 3w - 40 = 0.
3. Factor: two numbers multiplying to -40 and adding to 3 are 8 and -5, so (w + 8)(w - 5) = 0.
4. Roots: w = -8 or w = 5. A width cannot be negative, so w = 5.

The rectangle is 5 meters by 8 meters. Check: 8 is indeed 3 more than 5, and 5 × 8 = 40. Discarding the impossible root is not cheating; the algebra finds every number that fits the equation, and the situation then selects which fits reality.

Real-world applications

• Motion: flight times of balls, rockets, and divers come from setting height polynomials equal to zero.
• Design: finding dimensions that hit a target area or volume leads to quadratics, as in the rectangle example.
• Business: profit expressions equal zero exactly at the break-even production levels, and those levels are the roots of a quadratic when price affects sales.

Common misconceptions

• Factoring before reaching zero. The zero-product property needs one side to be exactly 0; rearrange first.
• Setting factors equal to the wrong number. From (x + 4)(x - 1) = 6 you may not write x + 4 = 6; the property works only for a product equal to zero.
• Dividing away the variable. Dividing x² = 7x by x silently discards the solution x = 0; factor instead.
• Reporting one root when there are two. Most quadratics have two solutions; find both, then let the context filter them.
• Sign errors reading roots from factors. The factor (x + 4) gives the root x = -4, not +4.

Recap

• A quadratic equation fits ax² + bx + c = 0 with a nonzero.
• Routine: move everything to one side, factor, set each factor to zero, solve, check.
• The zero-product property is the engine: a zero product forces a zero factor.
• Factor out a lone variable rather than dividing by it.
• In word problems, keep the root that makes physical sense.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 7: Factoring, section "Quadratic Equations" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Quadratic functions and equations" unit: solving by factoring (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Quadratic Equations, Part I" (tutorial.math.lamar.edu).

The parabola

The graph of a quadratic function y = ax² + bx + c is a smooth U-shaped curve called a parabola. If a is positive, the parabola opens upward like a cup; if a is negative, it opens downward like a dome. The sign of a is the first thing to read off, because it tells you whether the curve has a lowest point or a highest point.

The turning point of the parabola is its vertex - the lowest point when it opens up, or the highest point when it opens down. A vertical line through the vertex, called the axis of symmetry, splits the parabola into two mirror-image halves. This symmetry means every point on one side has a twin at the same height on the other side.

Where the graph meets the x-axis

The points where the parabola crosses the x-axis are the x-intercepts, and their x-values are exactly the solutions of ax² + bx + c = 0. This is why the factoring you learned finds the same numbers you would see on the graph. A parabola can cross the x-axis twice, touch it once, or miss it entirely, which corresponds to two, one, or no real solutions. The graph and the algebra are two views of the same fact.

The quadratic formula

Not every quadratic factors with whole numbers. The quadratic formula solves any quadratic in standard form ax² + bx + c = 0:

x = ( -b ± √(b² - 4ac) ) / (2a)

The expression under the square root, b² - 4ac, is the discriminant. If it is positive there are two real solutions; if it is zero there is exactly one (a double root); if it is negative there are no real solutions. Reading the discriminant first tells you how many answers to expect before you finish computing.

Worked example 1: use the quadratic formula

Solve x² + 3x - 4 = 0 with the formula, where a = 1, b = 3, c = -4.

1. Discriminant: b² - 4ac = 3² - 4(1)(-4) = 9 + 16 = 25.
2. Square root: √25 = 5.
3. Apply the formula: x = (-3 ± 5) / 2.
4. Two solutions: x = (-3 + 5)/2 = 1 and x = (-3 - 5)/2 = -4.

The solutions are 1 and -4, which match the factors (x - 1)(x + 4). The formula and factoring agree, as they must.

Worked example 2: an answer that will not factor

Solve x² - 4x + 1 = 0, where a = 1, b = -4, c = 1. Discriminant: (-4)² - 4(1)(1) = 16 - 4 = 12, which is positive but not a perfect square. So x = (4 ± √12) / 2. Since √12 = 2√3, this simplifies to x = 2 ± √3. There are two irrational solutions, roughly 3.73 and 0.27. This equation cannot be factored with whole numbers, which is exactly when the formula earns its keep.

Finding the vertex

The axis of symmetry sits at x = -b/(2a), the value halfway between the two roots. For y = x² - 4x + 1, that is x = 4/2 = 2. Substituting back gives the y-value of the vertex: y = 2² - 4(2) + 1 = -3, so the vertex is (2, -3), the lowest point of this upward parabola.

Why both tools matter

Factoring is fast when the numbers are friendly; the quadratic formula never fails, even when the answers are irrational or the expression will not factor neatly. Together they let you solve any quadratic that arises, and reading a parabola's intercepts and vertex gives you a picture of what those solutions mean, such as the maximum height of a projectile or the price that maximizes profit.

Worked example 3: the formula with a leading coefficient of 2

Solve 2x2 + 5x - 3 = 0, where a = 2, b = 5, c = -3.

1. Discriminant: b2 - 4ac = 52 - 4(2)(-3) = 25 + 24 = 49.
2. Square root: √49 = 7. Two real solutions are coming.
3. Formula: x = (-5 ± 7) / (2 · 2) = (-5 ± 7)/4.
4. Plus branch: x = (-5 + 7)/4 = 2/4 = 1/2. Minus branch: x = (-5 - 7)/4 = -12/4 = -3.

Check by factoring: (2x - 1)(x + 3) = 2x2 + 6x - x - 3 = 2x2 + 5x - 3, whose roots are indeed 1/2 and -3. Note the denominator 2a = 4 divides the entire numerator, both the -5 and the 7.

Worked example 4: maximum height of a projectile

A rocket's height in feet is h = -16t2 + 64t + 5. Find its maximum height.

1. The parabola opens downward (a = -16 is negative), so the vertex is the peak.
2. Vertex time: t = -b/(2a) = -64 / (2 × (-16)) = -64 / (-32) = 2 seconds.
3. Height at t = 2: h = -16(2)2 + 64(2) + 5 = -16(4) + 128 + 5 = -64 + 128 + 5 = 69 feet.

The rocket peaks at 69 feet after 2 seconds. The 5 in the equation is the launch height, the y-intercept of the story. Symmetry adds a bonus fact for free: the rocket passes 5 feet again at t = 4, twice the vertex time.

Real-world applications

• Trajectories: balls, fountains, and fireworks trace parabolas; vertex questions are "how high," and root questions are "when does it land."
• Design: satellite dishes, car headlights, and suspension bridge cables use parabolic shapes for their reflective and load-spreading properties.
• Business: revenue versus price is often a downward parabola; the vertex names the price that maximizes revenue.
• Safety engineering: stopping-distance formulas grow with the square of speed, so doubling speed roughly quadruples braking distance.

Common misconceptions

• Using b instead of -b. If b = -4, the formula starts with -(-4) = +4; track the sign change.
• Dividing only the square root by 2a. The entire numerator, -b and the root together, sits over 2a.
• Reading a, b, c from an unarranged equation. Move everything to one side equal to zero before naming the coefficients.
• Keeping only the plus branch. The ± produces two answers whenever the discriminant is positive.
• Concluding "no factoring means no solutions." Irrational roots like 2 ± √3 are perfectly real; only a negative discriminant means no real solutions.

Recap

• Quadratic graphs are parabolas: upward when a is positive, downward when a is negative.
• The vertex sits on the axis of symmetry x = -b/(2a); substitute back for its height.
• The quadratic formula solves any quadratic in standard form.
• The discriminant b² - 4ac predicts two, one, or zero real solutions before you finish.
• Factoring is fastest when it works; the formula always works.

Sources

• OpenStax, Elementary Algebra 2e, Chapter 10: Quadratic Equations, sections "Solve Quadratic Equations Using the Quadratic Formula" and "Graph Quadratic Equations in Two Variables" (openstax.org/details/books/elementary-algebra-2e).
• Khan Academy, Algebra 1, "Quadratic functions and equations" unit (khanacademy.org/math/algebra).
• Paul Dawkins, Paul's Online Math Notes, Algebra, "Quadratic Equations, Part II" and "Parabolas" (tutorial.math.lamar.edu).