This worksheet provides 20 practice problems for solving quadratic equations, ranging from simple differences of squares to trinomials.

Let's solve each quadratic equation in the worksheet step by step. The general form of a quadratic equation is:

$$
ax^2 + bx + c = 0
$$

We can solve these equations using factoring, completing the square, or the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

But for most of these, factoring will work well.

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1. $ x^2 + 5x + 6 = 0 $

Factor:
Find two numbers that multiply to 6 and add to 5 → 2 and 3
$$
(x + 2)(x + 3) = 0 \Rightarrow x = -2, -3
$$

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2. $ x^2 - 36 = 0 $

Difference of squares:
$$
x^2 = 36 \Rightarrow x = \pm 6
$$

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3. $ x^2 - 2x - 15 = 0 $

Factor:
Numbers that multiply to -15 and add to -2 → -5 and 3
$$
(x - 5)(x + 3) = 0 \Rightarrow x = 5, -3
$$

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4. $ x^2 - 3x - 28 = 0 $

Factors of -28 that add to -3 → -7 and 4
$$
(x - 7)(x + 4) = 0 \Rightarrow x = 7, -4
$$

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5. $ x^2 + 11x + 18 = 0 $

Factors of 18 that add to 11 → 9 and 2
$$
(x + 9)(x + 2) = 0 \Rightarrow x = -9, -2
$$

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6. $ x^2 + 14x + 48 = 0 $

Factors of 48 that add to 14 → 6 and 8
$$
(x + 6)(x + 8) = 0 \Rightarrow x = -6, -8
$$

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7. $ x^2 + x - 2 = 0 $

Factors of -2 that add to 1 → 2 and -1
$$
(x + 2)(x - 1) = 0 \Rightarrow x = -2, 1
$$

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8. $ x^2 - 11x + 28 = 0 $

Factors of 28 that add to -11 → -7 and -4
$$
(x - 7)(x - 4) = 0 \Rightarrow x = 7, 4
$$

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9. $ x^2 - 6x - 27 = 0 $

Factors of -27 that add to -6 → -9 and 3
$$
(x - 9)(x + 3) = 0 \Rightarrow x = 9, -3
$$

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10. $ x^2 - 3x - 40 = 0 $

Factors of -40 that add to -3 → -8 and 5
$$
(x - 8)(x + 5) = 0 \Rightarrow x = 8, -5
$$

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11. $ x^2 - 2x + 1 = 0 $

Perfect square trinomial:
$$
(x - 1)^2 = 0 \Rightarrow x = 1 \text{ (double root)}
$$

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12. $ x^2 - 7x + 6 = 0 $

Factors of 6 that add to -7 → -6 and -1
$$
(x - 6)(x - 1) = 0 \Rightarrow x = 6, 1
$$

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13. $ x^2 - 2x - 35 = 0 $

Factors of -35 that add to -2 → -7 and 5
$$
(x - 7)(x + 5) = 0 \Rightarrow x = 7, -5
$$

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14. $ x^2 + 5x - 14 = 0 $

Factors of -14 that add to 5 → 7 and -2
$$
(x + 7)(x - 2) = 0 \Rightarrow x = -7, 2
$$

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15. $ x^2 - x - 30 = 0 $

Factors of -30 that add to -1 → -6 and 5
$$
(x - 6)(x + 5) = 0 \Rightarrow x = 6, -5
$$

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16. $ x^2 - 14x + 45 = 0 $

Factors of 45 that add to -14 → -9 and -5
$$
(x - 9)(x - 5) = 0 \Rightarrow x = 9, 5
$$

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17. $ x^2 + 9x + 18 = 0 $

Factors of 18 that add to 9 → 6 and 3
$$
(x + 6)(x + 3) = 0 \Rightarrow x = -6, -3
$$

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18. $ x^2 - 14x + 48 = 0 $

Factors of 48 that add to -14 → -8 and -6
$$
(x - 8)(x - 6) = 0 \Rightarrow x = 8, 6
$$

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19. $ x^2 + 12x + 27 = 0 $

Factors of 27 that add to 12 → 9 and 3
$$
(x + 9)(x + 3) = 0 \Rightarrow x = -9, -3
$$

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20. $ x^2 + 2x - 3 = 0 $

Factors of -3 that add to 2 → 3 and -1
$$
(x + 3)(x - 1) = 0 \Rightarrow x = -3, 1
$$

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✔ Final Answers:

| Problem | Solutions |
|--------|-----------|
| 1 | $ x = -2, -3 $ |
| 2 | $ x = -6, 6 $ |
| 3 | $ x = -3, 5 $ |
| 4 | $ x = -4, 7 $ |
| 5 | $ x = -9, -2 $ |
| 6 | $ x = -8, -6 $ |
| 7 | $ x = -2, 1 $ |
| 8 | $ x = 4, 7 $ |
| 9 | $ x = -3, 9 $ |
| 10 | $ x = -5, 8 $ |
| 11 | $ x = 1 $ (double root) |
| 12 | $ x = 1, 6 $ |
| 13 | $ x = -5, 7 $ |
| 14 | $ x = -7, 2 $ |
| 15 | $ x = -5, 6 $ |
| 16 | $ x = 5, 9 $ |
| 17 | $ x = -6, -3 $ |
| 18 | $ x = 6, 8 $ |
| 19 | $ x = -9, -3 $ |
| 20 | $ x = -3, 1 $ |

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✔ Summary: All equations were solved by factoring. Most are factorable with integer roots. For any that aren't easily factorable, the quadratic formula could be used, but all here factor nicely.

Let me know if you'd like the steps shown in a different format (e.g., with diagrams or tables)!