Clean, printable algebra worksheet featuring twenty quadratic expression problems arranged in two columns.

The worksheet asks you to factor each quadratic expression. Factoring a quadratic expression means writing it as a product of two binomials.

We'll go through each quadratic expression one by one, factoring them using the AC method or by finding two numbers that multiply to the constant term and add to the coefficient of the middle term (for expressions in the form $ x^2 + bx + c $).

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✔ Step-by-step Solutions

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1. $ x^2 + 6x - 7 $
Find two numbers that multiply to $-7$ and add to $6$:
→ $7$ and $-1$: $7 \times (-1) = -7$, $7 + (-1) = 6$
→ $ (x + 7)(x - 1) $

2. $ x^2 - 2x - 15 $
Numbers: $-5$ and $3$: $(-5)(3) = -15$, $-5 + 3 = -2$
→ $ (x - 5)(x + 3) $

3. $ x^2 - 9x + 14 $
Numbers: $-7$ and $-2$: $(-7)(-2) = 14$, $-7 + (-2) = -9$
→ $ (x - 7)(x - 2) $

4. $ x^2 - 8x + 12 $
Numbers: $-6$ and $-2$: $(-6)(-2) = 12$, $-6 + (-2) = -8$
→ $ (x - 6)(x - 2) $

5. $ x^2 + 17x + 72 $
Numbers: $8$ and $9$: $8 \times 9 = 72$, $8 + 9 = 17$
→ $ (x + 8)(x + 9) $

6. $ x^2 - 2x - 24 $
Numbers: $-6$ and $4$: $(-6)(4) = -24$, $-6 + 4 = -2$
→ $ (x - 6)(x + 4) $

7. $ x^2 + 3x - 18 $
Numbers: $6$ and $-3$: $6 \times (-3) = -18$, $6 + (-3) = 3$
→ $ (x + 6)(x - 3) $

8. $ x^2 + x - 72 $
Numbers: $9$ and $-8$: $9 \times (-8) = -72$, $9 + (-8) = 1$
→ $ (x + 9)(x - 8) $

9. $ x^2 + 5x + 6 $
Numbers: $2$ and $3$: $2 \times 3 = 6$, $2 + 3 = 5$
→ $ (x + 2)(x + 3) $

10. $ x^2 - 12x + 35 $
Numbers: $-7$ and $-5$: $(-7)(-5) = 35$, $-7 + (-5) = -12$
→ $ (x - 7)(x - 5) $

11. $ x^2 - 4x - 12 $
Numbers: $-6$ and $2$: $(-6)(2) = -12$, $-6 + 2 = -4$
→ $ (x - 6)(x + 2) $

12. $ x^2 + 10x + 16 $
Numbers: $2$ and $8$: $2 \times 8 = 16$, $2 + 8 = 10$
→ $ (x + 2)(x + 8) $

13. $ x^2 + 2x - 8 $
Numbers: $4$ and $-2$: $4 \times (-2) = -8$, $4 + (-2) = 2$
→ $ (x + 4)(x - 2) $

14. $ x^2 + 14x + 48 $
Numbers: $6$ and $8$: $6 \times 8 = 48$, $6 + 8 = 14$
→ $ (x + 6)(x + 8) $

15. $ x^2 + 8x + 12 $
Numbers: $2$ and $6$: $2 \times 6 = 12$, $2 + 6 = 8$
→ $ (x + 2)(x + 6) $

16. $ x^2 - 3x - 18 $
Numbers: $-6$ and $3$: $(-6)(3) = -18$, $-6 + 3 = -3$
→ $ (x - 6)(x + 3) $

17. $ x^2 - 13x + 42 $
Numbers: $-6$ and $-7$: $(-6)(-7) = 42$, $-6 + (-7) = -13$
→ $ (x - 6)(x - 7) $

18. $ x^2 - 7x - 8 $
Numbers: $-8$ and $1$: $(-8)(1) = -8$, $-8 + 1 = -7$
→ $ (x - 8)(x + 1) $

19. $ x^2 - 2x - 8 $
Numbers: $-4$ and $2$: $(-4)(2) = -8$, $-4 + 2 = -2$
→ $ (x - 4)(x + 2) $

20. $ x^2 + 8x + 7 $
Numbers: $1$ and $7$: $1 \times 7 = 7$, $1 + 7 = 8$
→ $ (x + 1)(x + 7) $

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✔ Final Answers (Factored Form):

| Original Expression | Factored Form |
|---------------------|---------------|
| $ x^2 + 6x - 7 $ | $ (x + 7)(x - 1) $ |
| $ x^2 - 2x - 15 $ | $ (x - 5)(x + 3) $ |
| $ x^2 - 9x + 14 $ | $ (x - 7)(x - 2) $ |
| $ x^2 - 8x + 12 $ | $ (x - 6)(x - 2) $ |
| $ x^2 + 17x + 72 $ | $ (x + 8)(x + 9) $ |
| $ x^2 - 2x - 24 $ | $ (x - 6)(x + 4) $ |
| $ x^2 + 3x - 18 $ | $ (x + 6)(x - 3) $ |
| $ x^2 + x - 72 $ | $ (x + 9)(x - 8) $ |
| $ x^2 + 5x + 6 $ | $ (x + 2)(x + 3) $ |
| $ x^2 - 12x + 35 $ | $ (x - 7)(x - 5) $ |
| $ x^2 - 4x - 12 $ | $ (x - 6)(x + 2) $ |
| $ x^2 + 10x + 16 $ | $ (x + 2)(x + 8) $ |
| $ x^2 + 2x - 8 $ | $ (x + 4)(x - 2) $ |
| $ x^2 + 14x + 48 $ | $ (x + 6)(x + 8) $ |
| $ x^2 + 8x + 12 $ | $ (x + 2)(x + 6) $ |
| $ x^2 - 3x - 18 $ | $ (x - 6)(x + 3) $ |
| $ x^2 - 13x + 42 $ | $ (x - 6)(x - 7) $ |
| $ x^2 - 7x - 8 $ | $ (x - 8)(x + 1) $ |
| $ x^2 - 2x - 8 $ | $ (x - 4)(x + 2) $ |
| $ x^2 + 8x + 7 $ | $ (x + 1)(x + 7) $ |

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🔍 Explanation:

To factor a quadratic like $ x^2 + bx + c $:
1. Look for two numbers that:
- Multiply to $c$
- Add to $b$
2. Write the expression as $ (x + m)(x + n) $, where $m$ and $n$ are those numbers.

This method works when the leading coefficient is 1 (i.e., no coefficient on $x^2$). For more complex quadratics (like $ ax^2 + bx + c $ with $a \neq 1$), we'd use the AC method, but all these are monic (leading coefficient = 1), so this approach suffices.

Let me know if you want to see how to solve any of these using the quadratic formula instead!