Quadratics: factoring and solving

A quadratic is an equation with x²

How to factor and solve them

1. 1

   Spot a quadratic and put it in standard form

   ax² + bx + c = 0

   This is standard form: the x² term first, then the x term, then the plain number, and 0 alone on the other side. The letters a, b and c are just the numbers in those places. In this lesson a is almost always 1, so you will see x² + bx + c.

   Before you factor or solve, move every term to one side. When a term moves across the equals sign, its sign changes: + becomes −, and − becomes +.

   Already in standard form

   x² − 6x + 5 = 0

   a = 1, b = −6, c = 5

   Move a term first

   x² + 18 = 9x

   The 9x moves left and becomes −9x: x² − 9x + 18 = 0

   Not a quadratic

   4x + 7 = 15

   No x², so it is a linear equation

   Next step: Factor x² + bx + c ↓
2. 2

   Factor x² + bx + c: multiply to c, add to b

   Factoring means writing an expression as a multiplication. For x² + bx + c you want two sets of parentheses, (x + m)(x + n). The two numbers m and n must multiply to c and add to b.

   List the pairs of numbers that multiply to c, then pick the pair with the right sum. The signs tell you where to look:

   Reading the signs

   If c is…	and b is…	the two numbers are
   positive	positive	both positive
   positive	negative	both negative
   negative	either	one positive, one negative; the one that is bigger without its sign takes the sign of b

   Example

   Factor x² − 11x + 24

   1. c = 24 is positive and b = −11 is negative, so both numbers are negative.
   2. Pairs that multiply to 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6.
   3. Only one pair adds up to 11: 3 + 8 = 11. Make them negative: −3 and −8.
   4. x² − 11x + 24 = (x − 3)(x − 8).

   Answer (x − 3)(x − 8)

   Example

   Factor x² + 4x − 21

   1. c = −21 is negative, so one number is positive and one is negative.
   2. Pairs for 21: 1 and 21, 3 and 7. With opposite signs, the sum is the difference: 7 − 3 = 4.
   3. b is positive, so the number that is bigger without its sign (7) is positive: +7 and −3.
   4. (x + 7)(x − 3). Check the middle: 7x − 3x = 4x.

   Answer (x + 7)(x − 3)

   Next step: a² − b² ↓
3. 3

   Difference of squares: a² − b² = (a + b)(a − b)

   a² − b² = (a + b)(a − b)

   A perfect square is a number times itself, like 36 = 6 × 6. When a perfect square is minus another perfect square, with no middle term, write the two square roots twice: once with +, once with −. When you multiply back, the middle terms cancel.

   It only works with a minus. A sum of squares, like x² + 36, does not factor.

   Know the perfect squares by heart: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.

   Plain

   x² − 64

   64 = 8², so (x + 8)(x − 8)

   A number in front

   9x² − 25

   9x² = (3x)² and 25 = 5², so (3x + 5)(3x − 5)

   A sum

   x² + 36

   Does not factor

   Shortcut

   Mental math with it

   21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399. When two numbers sit the same distance from a round number, square the round number and subtract the square of that distance. For 22 × 18 the distance is 2: 20² − 2² = 400 − 4 = 396.

   Next step: Zero product ↓
4. 4

   Solve by factoring: the zero product rule

   If two numbers multiply to 0, at least one of them is 0. There is no other way to get 0. So once the equation reads (…)(…) = 0, set each set of parentheses equal to 0 and solve each small equation.

   This only works with 0 on the other side. That is why you move every term first. And watch the signs: the parentheses (x + 6) give x = −6, not 6.

   1. x² + 2x = 24
   2. x² + 2x − 24 = 0
   3. (x + 6)(x − 4) = 0
   4. x + 6 = 0 or x − 4 = 0
   5. x = −6 or x = 4

   Example

   Solve x² = 7x

   1. Do not divide both sides by x. You would lose a solution.
   2. Move 7x across: x² − 7x = 0.
   3. Both terms have an x, so take it out: x(x − 7) = 0.
   4. x = 0 or x − 7 = 0, so x = 0 or x = 7.

   Answer x = 0 or x = 7

   Next step: Check ↓
5. 5

   Check each root in the original equation

   A root is right if it makes the equation true. Put it in place of every x and do the arithmetic. If the two sides match, it works. It takes a few seconds and catches sign mistakes.

   For a negative root, keep the parentheses: (−4)² is 16, not −16.

   Here is the equation from step 1, x² − 9x + 18 = 0, which factors as (x − 3)(x − 6) = 0.

   1. x² − 9x + 18, with x = 3
   2. (3)² − 9(3) + 18
   3. 9 − 27 + 18
   4. 0

   1. x² − 9x + 18, with x = 6
   2. (6)² − 9(6) + 18
   3. 36 − 54 + 18
   4. 0

   Next step: Test the choices ↓
6. 6

   Plug in the answer choices

   On a multiple-choice test, the answer is already on the screen. If factoring is slow, put each choice into the equation and keep the one that gives 0. Start with the choice that is easiest to compute, and stop as soon as one works.

   Shortcut

   The roots add to −b and multiply to c

   In x² + bx + c = 0, the two roots always add up to −b and multiply to c. For x² − 9x + 18 = 0, b is −9, so −b is 9: the roots 3 and 6 add to 9 and multiply to 18. Use it to check a pair of roots at a glance.

   Example

   Which is a solution of x² − 13x + 42 = 0? (A) 2 (B) 6 (C) 13 (D) 42

   1. Be suspicious of 13 and 42: they are just numbers copied from the equation.
   2. Try 2 first, it is the easiest: 2² − 13 × 2 + 42 = 4 − 26 + 42 = 20. Not 0, so 2 is out.
   3. Try 6: 6² − 13 × 6 + 42 = 36 − 78 + 42 = 0. It works.
   4. The answer is 6. The other root is 7: 6 + 7 = 13 and 6 × 7 = 42.

   Answer (B) 6

   Next step: Three examples ↓
7. 7

   Three worked examples, from easy to test level

   Each one starts with a first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

   The test-level one is a word problem. Write the equation first, then factor, then ask which root the story allows, and what the question really asks for.

   Example · Easy

   Solve x² − 121 = 0.

   First try(x − 11)(x − 11) = 0, so x = 11

   1. Multiply that try back: (x − 11)(x − 11) = x² − 22x + 121. It has a middle term, so it is wrong.
   2. x² − 121 is a difference of squares, because 121 = 11².
   3. So x² − 121 = (x + 11)(x − 11) = 0.
   4. x + 11 = 0 or x − 11 = 0, so x = −11 or x = 11.

   Answer x = −11 or x = 11

   Example · Medium

   Solve x² + x − 42 = 0.

   First try(x + 7)(x − 6) = 0, so x = 7 or x = −6

   1. The x in the middle is 1x, so b = 1. The factoring is right: 7 × (−6) = −42 and 7 + (−6) = 1. The roots are wrong.
   2. Set each set of parentheses equal to 0. x + 7 = 0 means x = −7, not 7.
   3. x − 6 = 0 means x = 6.
   4. So x = −7 or x = 6. Check 6: 36 + 6 − 42 = 0.

   Answer x = −7 or x = 6

   Example · Test level

   Two barracks rooms have consecutive numbers (one right after the other). The two room numbers multiply to 132. What is the larger room number?

   First tryx(x + 1) = x² + 1 = 132, so x² = 131

   1. That try forgot to multiply x by the 1: x(x + 1) is x² + x, not x² + 1.
   2. Call the smaller number x. The larger one is x + 1. So x² + x = 132, and x² + x − 132 = 0.
   3. Two numbers that multiply to −132 and add to 1: 12 and −11. So (x + 12)(x − 11) = 0.
   4. x = −12 or x = 11. A room number is not negative, so the smaller room is 11.
   5. The question asks for the larger one: 11 + 1 = 12. Check: 11 × 12 = 132.

   Answer Room 12

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

12 flashcards

Tap a card to flip it. Keep the ones you miss until they stick.