Solve Quadratic Equations | Math Coach Amy

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Factor · Square root · Formula

x = −b ± √(b² − 4ac)2a

What to Remember

Factor

Write = 0, factor, then if AB = 0, A = 0 or B = 0.

Square root

If x² = k and k > 0, then x = ±√k. If k = 0, one solution. If k < 0, no real solution.

Quadratic formula

For ax² + bx + c = 0,

x = −b ± √(b² − 4ac)2a

Rational or irrational answers are both allowed.

A story can reject a root

Length, time, and counts cannot be negative in most stories. Check both algebra and the story.

Level 1

Easy

Factor or take square roots. Integer answers.

Example

Solve. x² − 7x + 12 = 0

(x − 3)(x − 4) = 0

x = 3 or x = 4

Check.

3² − 7(3) + 12 = 0

4² − 7(4) + 12 = 0

Answer: x = 3, x = 4

Solve. x² = 36

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x = −6 or x = 6

Answer: x = −6, x = 6

Solve. x² + 8x + 15 = 0

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(x + 3)(x + 5) = 0

x = −3 or x = −5

Answer: x = −5, x = −3

Solve. x² − 25 = 0

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(x + 5)(x − 5) = 0

x = −5 or x = 5

Answer: x = −5, x = 5

Solve. (x − 2)² = 0

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x = 2

One real solution (a double root).

Answer: x = 2

Solve. x(x − 9) = 0

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x = 0 or x = 9

Answer: x = 0, x = 9

Level 2

Medium

a > 1, a square-root leftover, or a short story.

Example

Solve. 2x² − 5x − 3 = 0

(2x + 1)(x − 3) = 0

2x + 1 = 0 or x − 3 = 0

x = −12 or x = 3

Check x = 3: 2(9) − 15 − 3 = 0.

Answer: x = −1/2, x = 3

Solve. x² = 20

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x = ±√20

√20 = 2√5

x = ±2√5

Answer: x = ±2√5

Solve. 3x² = 12x

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3x² − 12x = 0

3x(x − 4) = 0

x = 0 or x = 4

Answer: x = 0, x = 4

Solve. (x + 1)² = 16

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x + 1 = ±4

x = 3 or x = −5

Answer: x = −5, x = 3

The product of two consecutive integers is 72. Find them.

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n(n + 1) = 72

n² + n − 72 = 0

(n + 9)(n − 8) = 0

n = 8 or n = −9

The pairs are 8 and 9, or −9 and −8.

Answer: 8 and 9, or −9 and −8

Solve. x² + 2x − 8 = 0

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(x + 4)(x − 2) = 0

x = −4 or x = 2

Answer: x = −4, x = 2

Level 3

Hard

Quadratic formula with a simplified radical, or a story that drops a root.

Example

Solve. x² + 4x − 1 = 0

x = −b ± √(b² − 4ac)2a

a = 1, b = 4, c = −1

x = −4 ± √(16 − 4(1)(−1))2

x = −4 ± √(16 + 4)2

x = −4 ± √202

x = −4 ± 2√52

x = −2 ± √5

Answer: −2 ± √5

A rectangle’s length is 4 cm more than its width. The area is 45 cm². Find the width.

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w(w + 4) = 45

w² + 4w − 45 = 0

(w + 9)(w − 5) = 0

w = 5 or w = −9

Width cannot be negative, so w = 5 cm.

Length 9 cm. Check: 5(9) = 45.

Answer: 5 cm

Solve. 2x² + 3x − 4 = 0

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x = −b ± √(b² − 4ac)2a

x = −3 ± √(9 + 32)4

x = −3 ± √414

Answer: −3 ± √414

A ball’s height is h(t) = −16t² + 48t. When is it 32 feet high?

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−16t² + 48t = 32

−16t² + 48t − 32 = 0

Divide by −16.

t² − 3t + 2 = 0

(t − 1)(t − 2) = 0

t = 1 s or t = 2 s

Going up at 1 s, coming down at 2 s.

Answer: t = 1 s, t = 2 s

Solve. x² − 6x + 4 = 0

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x = 6 ± √(36 − 16)2

x = 6 ± √202

x = 6 ± 2√52

x = 3 ± √5

Answer: x = 3 ± √5

The square of a number is 12 more than 4 times the number. Find the number(s).

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n² = 4n + 12

n² − 4n − 12 = 0

(n − 6)(n + 2) = 0

n = 6 or n = −2

Both are numbers, so both solutions stay.

Answer: 6 and −2

Super Challenge

A garden is a square of side x with a 2-by-2 square cut from one corner. The remaining area is 45. Find x.

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x² − 4 = 45

x² = 49

x = 7 (side length > 2)

x = −7 is not a length.

Answer: x = 7

Solve (2x − 1)² = 18 two ways: square root, then expand and use the formula. Show the answers match.

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2x − 1 = ±√18 = ±3√2

x = 1 ± 3√22

Expand: 4x² − 4x + 1 − 18 = 0, 4x² − 4x − 17 = 0

x = 4 ± √(16 + 272)8 = 4 ± √2888

√288 = 12√2, so 4 ± 12√28 = 1 ± 3√22

Answer: 1 ± 3√22

Why can x² = 8 have two real solutions while a story “a square rug has area 8” has only one?

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Algebra: x = ±2√2.

A length is positive, so x = 2√2 only.

Answer: only the positive length

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