Factor Quadratics (a = 1) | Math Coach Amy

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Key Ideas Easy Medium Hard Super Challenge

What to Remember

GCF first

If every term shares a factor, factor it out before you look at the trinomial.

Multiply to the last, add to the middle

For x² + bx + c, find two numbers that multiply to c and add to b.

Check by multiplying

FOIL the factors. You should get the original trinomial.

Prime

If no integer pair works, the trinomial does not factor over the integers.

Level 1

Easy

a = 1. Positive factors.

Example

Factor. x² + 8x + 12

Find two numbers that multiply to 12 and add to 8.

2 and 6:

2*6 = 12 and 2+6 = 8.

(x+2)(x+6)

Answer: (x+2)(x+6)

Factor. x² + 8x + 15

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Find two numbers that multiply to 15 and add to 8.

3 and 5:

3*5 = 15 and 3+5 = 8.

(x+3)(x+5)

Answer: (x+3)(x+5)

Factor. x² + 9x + 18

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Find two numbers that multiply to 18 and add to 9.

3 and 6:

3*6 = 18 and 3+6 = 9.

(x+3)(x+6)

Answer: (x+3)(x+6)

Factor. x² + 9x + 14

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Find two numbers that multiply to 14 and add to 9.

2 and 7:

2*7 = 14 and 2+7 = 9.

(x+2)(x+7)

Answer: (x+2)(x+7)

Factor. x² + 10x + 21

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Find two numbers that multiply to 21 and add to 10.

3 and 7:

3*7 = 21 and 3+7 = 10.

(x+3)(x+7)

Answer: (x+3)(x+7)

Factor. 5x + 15

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GCF is 5.

5(x + 3)

Answer: 5(x + 3)

Level 2

Medium

A negative in c, or GCF then a = 1.

Example

Factor. x² − 2x − 24

Find two numbers that multiply to −24 and add to −2.

−6 and 4:

−6*4 = −24 and −6+4 = −2.

(x−6)(x+4)

Answer: (x−6)(x+4)

Factor. x² + 4x − 21

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Find two numbers that multiply to −21 and add to 4.

−3 and 7:

−3*7 = −21 and −3+7 = 4.

(x−3)(x+7)

Answer: (x−3)(x+7)

Factor. x² − 6x − 16

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Find two numbers that multiply to −16 and add to −6.

−8 and 2:

−8*2 = −16 and −8+2 = −6.

(x−8)(x+2)

Answer: (x−8)(x+2)

Factor. 2x² + 10x + 12

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GCF is 2.

2(x² + 5x + 6)

Find two numbers that multiply to 6 and add to 5.

2 and 3.

2(x+2)(x+3)

Answer: 2(x+2)(x+3)

Factor. x² − 16

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Difference of squares.

x² − 16 = (x + 4)(x − 4)

Answer: (x + 4)(x − 4)

Factor. x² − 12x + 36

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Find two numbers that multiply to 36 and add to −12.

−6 and −6:

−6*−6 = 36 and −6+(−6) = −12.

(x−6)(x−6)

Answer: (x−6)(x−6)

Level 3

Hard

Larger c, or GCF with a difference of squares.

Example

Factor. x² + 2x − 48

Find two numbers that multiply to −48 and add to 2.

−6 and 8:

−6*8 = −48 and −6+8 = 2.

(x−6)(x+8)

Answer: (x−6)(x+8)

Factor. x² − 13x + 40

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Find two numbers that multiply to 40 and add to −13.

−5 and −8:

−5*−8 = 40 and −5+(−8) = −13.

(x−5)(x−8)

Answer: (x−5)(x−8)

Factor completely. 3x² − 3x − 18

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GCF is 3.

3(x² − x − 6)

Find two numbers that multiply to −6 and add to −1.

−3 and 2.

3(x−3)(x+2)

Answer: 3(x−3)(x+2)

Factor completely. 2x³ − 32x

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GCF = 2x

2x(x² − 16)

2x(x + 4)(x − 4)

Answer: 2x(x + 4)(x − 4)

Factor. x² + 2x + 5

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Two integers that multiply to 5 and add to 2: none.

Prime over the integers.

Answer: prime

Factor. x² − 4x − 45

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Find two numbers that multiply to −45 and add to −4.

−9 and 5:

−9*5 = −45 and −9+5 = −4.

(x−9)(x+5)

Answer: (x−9)(x+5)

Super Challenge

Find all integer k so that x² + kx + 20 factors over the integers.

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Pairs that multiply to 20: ±1 and ±20, ±2 and ±10, ±4 and ±5.

Sums: 21, −21, 19, −19, 12, −12, 8, −8, 9, −9, −3, 3.

Answer: k = ±21, ±19, ±12, ±9, ±8, ±3

A rectangle has area x² + 9x + 18. Write possible side lengths as binomials.

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(x + 3)(x + 6)

Answer: (x + 3)(x + 6)

Why does “multiply to c, add to b” fail if you forget the GCF in 2x² + 8x + 6?

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You would hunt for a pair that multiplies to 6 and adds to 8: none.

Factor 2 first: 2(x² + 4x + 3) = 2(x + 1)(x + 3).

Answer: 2(x + 1)(x + 3)

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