Factor Quadratics (a = 1) | Math Coach Amy

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Key Ideas Easy 1 Easy 2 Easy 3 Medium Hard

What to Remember

Multiply and add

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

x² + 5x + 6

What multiplies to 6 and adds to 5.

1*6

→ 2*3

(x+2)(x+3)

Prime

If no integer pair multiplies to 𝑐 and adds to 𝑏, the trinomial is prime.

x² + 5x + 3 is prime

GCF first on hard problems

Factor out the greatest common factor, then factor the leftover x² + 𝑏x + 𝑐 trinomial.

2x² − 8x − 10 = 2(x² − 4x − 5) = 2(x−5)(x+1)

Level 1

Easy · Group 1

Example

Factor. x² + 5x + 6

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

2 and 3: 2*3 = 6 and 2+3 = 5.

(x+2)(x+3)

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

Factor. x² + 7x + 12

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Find two numbers that multiply to 12 and add to 7.

3 and 4: 3*4 = 12 and 3+4 = 7.

(x+3)(x+4)

Answer: (x+3)(x+4)

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 + 8

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Find two numbers that multiply to 8 and add to 9.

1 and 8: 1*8 = 8 and 1+8 = 9.

(x+1)(x+8)

Answer: (x+1)(x+8)

Factor. x² + 6x + 8

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Find two numbers that multiply to 8 and add to 6.

2 and 4: 2*4 = 8 and 2+4 = 6.

(x+2)(x+4)

Answer: (x+2)(x+4)

Factor. x² + 4x + 6

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Find two integers that multiply to 6 and add to 4.

No integer pair works. This trinomial is prime.

Answer: prime

Level 1

Easy · Group 2

Example

Factor. x² − 7x + 10

Find two numbers that multiply to 10 and add to −7.

−2 and −5: −2*−5 = 10 and −2+(−5) = −7.

(x−2)(x−5)

Answer: (x−2)(x−5)

Factor. x² − 8x + 12

Show Answer

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 − 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 − 6x + 5

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

−1 and −5: −1*−5 = 5 and −1+(−5) = −6.

(x−1)(x−5)

Answer: (x−1)(x−5)

Factor. x² − 5x + 4

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Find two numbers that multiply to 4 and add to −5.

−1 and −4: −1*−4 = 4 and −1+(−4) = −5.

(x−1)(x−4)

Answer: (x−1)(x−4)

Factor. x² − 4x + 6

Show Answer

Find two integers that multiply to 6 and add to −4.

No integer pair works. This trinomial is prime.

Answer: prime

Level 1

Easy · Group 3

Example

Factor. x² + 2x − 15

Find two numbers that multiply to −15 and add to 2.

5 and −3: 5*−3 = −15 and 5+(−3) = 2.

(x+5)(x−3)

Answer: (x+5)(x−3)

Factor. x² + 3x − 10

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Find two numbers that multiply to −10 and add to 3.

5 and −2: 5*−2 = −10 and 5+(−2) = 3.

(x+5)(x−2)

Answer: (x+5)(x−2)

Factor. x² − 2x − 8

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Find two numbers that multiply to −8 and add to −2.

−4 and 2: −4*2 = −8 and −4+2 = −2.

(x−4)(x+2)

Answer: (x−4)(x+2)

Factor. x² + 4x − 12

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Find two numbers that multiply to −12 and add to 4.

6 and −2: 6*−2 = −12 and 6+(−2) = 4.

(x+6)(x−2)

Answer: (x+6)(x−2)

Factor. x² − 3x − 10

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Find two numbers that multiply to −10 and add to −3.

−5 and 2: −5*2 = −10 and −5+2 = −3.

(x−5)(x+2)

Answer: (x−5)(x+2)

Factor. x² + 2x − 6

Show Answer

Find two integers that multiply to −6 and add to 2.

No integer pair works. This trinomial is prime.

Answer: prime

Level 2

Medium

Example

Factor. x² + 13x + 36

Find two numbers that multiply to 36 and add to 13.

4 and 9: 4*9 = 36 and 4+9 = 13.

(x+4)(x+9)

Answer: (x+4)(x+9)

Factor. x² + 17x + 60

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Find two numbers that multiply to 60 and add to 17.

5 and 12: 5*12 = 60 and 5+12 = 17.

(x+5)(x+12)

Answer: (x+5)(x+12)

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. x² + 9x − 36

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Find two numbers that multiply to −36 and add to 9.

12 and −3: 12*−3 = −36 and 12+(−3) = 9.

(x+12)(x−3)

Answer: (x+12)(x−3)

Factor. x² − 11x − 42

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Find two numbers that multiply to −42 and add to −11.

−14 and 3: −14*3 = −42 and −14+3 = −11.

(x−14)(x+3)

Answer: (x−14)(x+3)

Factor. x² + 19x + 90

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Find two numbers that multiply to 90 and add to 19.

9 and 10: 9*10 = 90 and 9+10 = 19.

(x+9)(x+10)

Answer: (x+9)(x+10)

Factor. x² − 16x + 63

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Find two numbers that multiply to 63 and add to −16.

−7 and −9: −7*−9 = 63 and −7+(−9) = −16.

(x−7)(x−9)

Answer: (x−7)(x−9)

Factor. x² + 4x − 77

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Find two numbers that multiply to −77 and add to 4.

11 and −7: 11*−7 = −77 and 11+(−7) = 4.

(x+11)(x−7)

Answer: (x+11)(x−7)

Factor. x² − 5x − 84

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Find two numbers that multiply to −84 and add to −5.

−12 and 7: −12*7 = −84 and −12+7 = −5.

(x−12)(x+7)

Answer: (x−12)(x+7)

Factor. x² + 15x − 54

Show Answer

Find two numbers that multiply to −54 and add to 15.

18 and −3: 18*−3 = −54 and 18+(−3) = 15.

(x+18)(x−3)

Answer: (x+18)(x−3)

Factor. x² − 19x + 90

Show Answer

Find two numbers that multiply to 90 and add to −19.

−9 and −10: −9*−10 = 90 and −9+(−10) = −19.

(x−9)(x−10)

Answer: (x−9)(x−10)

Factor. x² + 11x + 15

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Find two integers that multiply to 15 and add to 11.

No integer pair works. This trinomial is prime.

Answer: prime

Factor. x² + 5x − 8

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Find two integers that multiply to −8 and add to 5.

No integer pair works. This trinomial is prime.

Answer: prime

Level 3

Hard

Factor out a GCF first. Then factor the leftover trinomial.

Example

Factor. 2x² − 8x − 10

GCF is 2: 2(x² − 4x − 5).

Find two numbers that multiply to −5 and add to −4: −5 and 1.

2(x−5)(x+1)

Answer: 2(x−5)(x+1)

Factor. 2x² + 12x + 10

Show Answer

GCF is 2: 2(x² + 6x + 5).

Find two numbers that multiply to 5 and add to 6: 1 and 5.

2(x+1)(x+5)

Answer: 2(x+1)(x+5)

Factor. 3x² − 15x + 18

Show Answer

GCF is 3: 3(x² − 5x + 6).

Find two numbers that multiply to 6 and add to −5: −2 and −3.

3(x−2)(x−3)

Answer: 3(x−2)(x−3)

Factor. 5x² + 5x − 30

Show Answer

GCF is 5: 5(x² + x − 6).

Find two numbers that multiply to −6 and add to 1: 3 and −2.

5(x+3)(x−2)

Answer: 5(x+3)(x−2)

Factor. 4x² + 36x + 80

Show Answer

GCF is 4: 4(x² + 9x + 20).

Find two numbers that multiply to 20 and add to 9: 4 and 5.

4(x+4)(x+5)

Answer: 4(x+4)(x+5)

Factor. 2x² − 22x + 56

Show Answer

GCF is 2: 2(x² − 11x + 28).

Find two numbers that multiply to 28 and add to −11: −4 and −7.

2(x−4)(x−7)

Answer: 2(x−4)(x−7)

Factor. 6x² + 6x − 72

Show Answer

GCF is 6: 6(x² + x − 12).

Find two numbers that multiply to −12 and add to 1: 4 and −3.

6(x+4)(x−3)

Answer: 6(x+4)(x−3)

Factor. 3x² − 39x + 126

Show Answer

GCF is 3: 3(x² − 13x + 42).

Find two numbers that multiply to 42 and add to −13: −6 and −7.

3(x−6)(x−7)

Answer: 3(x−6)(x−7)

Factor. 5x² + 35x − 90

Show Answer

GCF is 5: 5(x² + 7x − 18).

Find two numbers that multiply to −18 and add to 7: 9 and −2.

5(x+9)(x−2)

Answer: 5(x+9)(x−2)

Factor. 4x² − 4x − 120

Show Answer

GCF is 4: 4(x² − x − 30).

Find two numbers that multiply to −30 and add to −1: −6 and 5.

4(x−6)(x+5)

Answer: 4(x−6)(x+5)

Factor. 7x² + 77x + 210

Show Answer

GCF is 7: 7(x² + 11x + 30).

Find two numbers that multiply to 30 and add to 11: 5 and 6.

7(x+5)(x+6)

Answer: 7(x+5)(x+6)

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