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:

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

---

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

Step 1: Factor out the GCF (if any).
GCF of 2, -2, -24 is 2.

$$
2(x^2 - x - 12) = 0
$$

Now factor $ x^2 - x - 12 $:
We need two numbers that multiply to $-12$ and add to $-1$:
→ $-4$ and $3$

$$
x^2 - x - 12 = (x - 4)(x + 3)
$$

So:
$$
2(x - 4)(x + 3) = 0
\Rightarrow x = 4 \quad \text{or} \quad x = -3
$$

✔ Solutions: $ x = 4, -3 $

---

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

Find two numbers that multiply to $16$ and add to $10$:
→ $8$ and $2$

$$
x^2 + 10x + 16 = (x + 8)(x + 2) = 0
\Rightarrow x = -8 \quad \text{or} \quad x = -2
$$

✔ Solutions: $ x = -8, -2 $

---

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

Try factoring:
We need two numbers that multiply to $4 \times 9 = 36$, and add to $15$.
→ $12$ and $3$

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

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

✔ Solutions: $ x = -\frac{3}{4}, -3 $

---

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

Find two numbers that multiply to $42$ and add to $-13$:
→ $-6$ and $-7$

$$
x^2 - 13x + 42 = (x - 6)(x - 7) = 0
\Rightarrow x = 6 \quad \text{or} \quad x = 7
$$

✔ Solutions: $ x = 6, 7 $

---

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

Use AC method:
$ a = 6, b = -1, c = -40 $ → $ ac = 6 \times (-40) = -240 $

Find two numbers that multiply to $-240$ and add to $-1$:
→ $-16$ and $15$

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

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

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

---

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

Use AC method:
$ a = 2, b = 13, c = 15 $ → $ ac = 30 $

Find two numbers that multiply to $30$ and add to $13$:
→ $10$ and $3$

Break middle term:
$$
2x^2 + 10x + 3x + 15 = 0
\Rightarrow 2x(x + 5) + 3(x + 5) = 0
\Rightarrow (2x + 3)(x + 5) = 0
$$

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

✔ Solutions: $ x = -\frac{3}{2}, -5 $

---

✔ 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 $ |

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