Solving Quadratic Equations by Factoring worksheet - Free Printable

Let's solve each quadratic equation step by step. We'll use factoring where possible, and if factoring is difficult, we’ll use the quadratic formula:

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

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1. $ 2x^2 - 2x - 24 = 0 $

We can factor this.

First, factor out the GCF (if any):
No common factor in all terms.

Now look for two numbers that multiply to $ 2 \times (-24) = -48 $ and add to $-2$.

Try: $6$ and $-8$:
$6 \times (-8) = -48$, $6 + (-8) = -2$ ✔

So split the middle term:
$$
2x^2 + 6x - 8x - 24 = 0
$$
Group:
$$
(2x^2 + 6x) - (8x + 24) = 0 \\
2x(x + 3) - 8(x + 3) = 0 \\
(2x - 8)(x + 3) = 0
$$

Wait — actually, let's check:
$2x(x + 3) = 2x^2 + 6x$, and $-8(x + 3) = -8x -24$, so total: $2x^2 -2x -24$. Correct.

But we can simplify further:
Factor out 2 from first term:
$$
(2x - 8)(x + 3) = 0 \Rightarrow 2(x - 4)(x + 3) = 0
$$

So solutions:
$$
x = 4 \quad \text{or} \quad x = -3
$$

✔ Answer: $ x = 4, -3 $

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2. $ x^2 + 10x + 16 = 0 $

Look for two numbers that multiply to $16$ and add to $10$:
$8$ and $2$: $8 \times 2 = 16$, $8 + 2 = 10$

$$
(x + 8)(x + 2) = 0
$$

Solutions:
$$
x = -8 \quad \text{or} \quad x = -2
$$

✔ Answer: $ x = -8, -2 $

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3. $ 4x^2 + 15x + 9 = 0 $

Find two numbers that multiply to $4 \times 9 = 36$ and add to $15$.
Try: $12$ and $3$: $12 \times 3 = 36$, $12 + 3 = 15$ ✔

Split the middle term:
$$
4x^2 + 12x + 3x + 9 = 0 \\
(4x^2 + 12x) + (3x + 9) = 0 \\
4x(x + 3) + 3(x + 3) = 0 \\
(4x + 3)(x + 3) = 0
$$

Solutions:
$$
x = -\frac{3}{4} \quad \text{or} \quad x = -3
$$

✔ Answer: $ x = -\frac{3}{4}, -3 $

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4. $ x^2 - 13x + 42 = 0 $

Find two numbers that multiply to $42$ and add to $-13$:
Negative numbers: $-6$ and $-7$: $(-6)(-7)=42$, $-6 + (-7) = -13$ ✔

$$
(x - 6)(x - 7) = 0
$$

Solutions:
$$
x = 6 \quad \text{or} \quad x = 7
$$

✔ Answer: $ x = 6, 7 $

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5. $ 6x^2 - x - 40 = 0 $

Multiply $6 \times (-40) = -240$

Find two numbers that multiply to $-240$ and add to $-1$

Try: $-16$ and $15$: $-16 \times 15 = -240$, $-16 + 15 = -1$ ✔

Split the middle term:
$$
6x^2 - 16x + 15x - 40 = 0 \\
(6x^2 - 16x) + (15x - 40) = 0 \\
2x(3x - 8) + 5(3x - 8) = 0 \\
(2x + 5)(3x - 8) = 0
$$

Solutions:
$$
x = -\frac{5}{2} \quad \text{or} \quad x = \frac{8}{3}
$$

✔ Answer: $ x = -\frac{5}{2}, \frac{8}{3} $

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6. $ 2x^2 + 13x + 15 = 0 $

Multiply $2 \times 15 = 30$

Find two numbers that multiply to $30$ and add to $13$:
$10$ and $3$: $10 \times 3 = 30$, $10 + 3 = 13$ ✔

Split:
$$
2x^2 + 10x + 3x + 15 = 0 \\
(2x^2 + 10x) + (3x + 15) = 0 \\
2x(x + 5) + 3(x + 5) = 0 \\
(2x + 3)(x + 5) = 0
$$

Solutions:
$$
x = -\frac{3}{2} \quad \text{or} \quad x = -5
$$

✔ Answer: $ x = -\frac{3}{2}, -5 $

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✔ Final Answers Summary:

| Equation | Solutions |
|--------|-----------|
| $2x^2 - 2x - 24 = 0$ | $x = 4, -3$ |
| $x^2 + 10x + 16 = 0$ | $x = -8, -2$ |
| $4x^2 + 15x + 9 = 0$ | $x = -\frac{3}{4}, -3$ |
| $x^2 - 13x + 42 = 0$ | $x = 6, 7$ |
| $6x^2 - x - 40 = 0$ | $x = -\frac{5}{2}, \frac{8}{3}$ |
| $2x^2 + 13x + 15 = 0$ | $x = -\frac{3}{2}, -5$ |

All equations were solved using factoring, which works well when the trinomial factors nicely. If factoring is not possible, you would use the quadratic formula, but here all factor cleanly.

Let me know if you'd like a visual explanation or graph!