Solving Quadratic Equations worksheets - Free Printable

Let's solve each of the quadratic equations step by step using factoring or the quadratic formula, depending on which method is most efficient.

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🔷 Quadratic Formula Reminder:

For a quadratic equation in the form $ ax^2 + bx + c = 0 $, the solutions are:

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

We'll use factoring when possible; otherwise, we'll apply the quadratic formula.

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✔ 1) $ x^2 + 13x + 12 = 0 $

Factor:
Find two numbers that multiply to 12 and add to 13:
→ $ 12 $ and $ 1 $

$$
(x + 12)(x + 1) = 0
$$

Solutions:
$$
x = -12 \quad \text{or} \quad x = -1
$$

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✔ 2) $ 3x^2 - 7x + 4 = 0 $

Try factoring:
We need two numbers that multiply to $ 3 \times 4 = 12 $ and add to $-7$.
→ $-3$ and $-4$

Now split the middle term:
$$
3x^2 - 3x - 4x + 4 = 0 \\
3x(x - 1) -4(x - 1) = 0 \\
(3x - 4)(x - 1) = 0
$$

Solutions:
$$
x = \frac{4}{3} \quad \text{or} \quad x = 1
$$

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✔ 3) $ x^2 + 5x + 4 = 0 $

Factor:
Numbers that multiply to 4 and add to 5: $ 4 $ and $ 1 $

$$
(x + 4)(x + 1) = 0
$$

Solutions:
$$
x = -4 \quad \text{or} \quad x = -1
$$

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✔ 4) $ 2x^2 + 4x + 3 = 0 $

Try factoring:
Check if it factors nicely.
Discriminant: $ b^2 - 4ac = 4^2 - 4(2)(3) = 16 - 24 = -8 < 0 $

No real solutions — complex roots. Use quadratic formula.

$$
x = \frac{-4 \pm \sqrt{(-8)}}{2(2)} = \frac{-4 \pm \sqrt{-8}}{4} = \frac{-4 \pm 2i\sqrt{2}}{4} = \frac{-2 \pm i\sqrt{2}}{2}
$$

Solutions:
$$
x = \frac{-2 + i\sqrt{2}}{2} \quad \text{or} \quad x = \frac{-2 - i\sqrt{2}}{2}
$$

(Note: If only real solutions are required, this has no real solution.)

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✔ 5) $ 3x^2 + 8x - 3 = 0 $

Try factoring:
Multiply $ 3 \times (-3) = -9 $. Find two numbers that multiply to $-9$ and add to $8$:
→ $ 9 $ and $-1$

Split:
$$
3x^2 + 9x - x - 3 = 0 \\
3x(x + 3) -1(x + 3) = 0 \\
(3x - 1)(x + 3) = 0
$$

Solutions:
$$
x = \frac{1}{3} \quad \text{or} \quad x = -3
$$

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✔ 6) $ x^2 + 10x - 24 = 0 $

Factor:
Two numbers that multiply to $-24$ and add to $10$:
→ $ 12 $ and $-2$

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

Solutions:
$$
x = -12 \quad \text{or} \quad x = 2
$$

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✔ 7) $ x^2 + 3x + 3 = 0 $

Discriminant: $ b^2 - 4ac = 9 - 12 = -3 < 0 $

No real solutions. Use quadratic formula:

$$
x = \frac{-3 \pm \sqrt{-3}}{2} = \frac{-3 \pm i\sqrt{3}}{2}
$$

Solutions:
$$
x = \frac{-3 + i\sqrt{3}}{2} \quad \text{or} \quad x = \frac{-3 - i\sqrt{3}}{2}
$$

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✔ 8) $ 4x^2 + 6x - 4 = 0 $

Divide entire equation by 2:
$ 2x^2 + 3x - 2 = 0 $

Factor:
Need numbers that multiply to $-4$, add to $3$: $ 4 $ and $-1$

$$
2x^2 + 4x - x - 2 = 0 \\
2x(x + 2) -1(x + 2) = 0 \\
(2x - 1)(x + 2) = 0
$$

Solutions:
$$
x = \frac{1}{2} \quad \text{or} \quad x = -2
$$

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✔ 9) $ 4x^2 + 8x = 0 $

Factor out common term:
$$
4x(x + 2) = 0
$$

Solutions:
$$
x = 0 \quad \text{or} \quad x = -2
$$

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✔ 10) $ 4x^2 - 24x + 36 = 0 $

Divide by 4:
$$
x^2 - 6x + 9 = 0
$$

This is a perfect square:
$$
(x - 3)^2 = 0
$$

Solution:
$$
x = 3 \quad \text{(double root)}
$$

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📌 Final Answers Summary:

| Problem | Solution(s) |
|--------|-------------|
| 1) | $ x = -12, -1 $ |
| 2) | $ x = \frac{4}{3}, 1 $ |
| 3) | $ x = -4, -1 $ |
| 4) | $ x = \frac{-2 \pm i\sqrt{2}}{2} $ (no real) |
| 5) | $ x = \frac{1}{3}, -3 $ |
| 6) | $ x = -12, 2 $ |
| 7) | $ x = \frac{-3 \pm i\sqrt{3}}{2} $ (no real) |
| 8) | $ x = \frac{1}{2}, -2 $ |
| 9) | $ x = 0, -2 $ |
| 10) | $ x = 3 $ (double root) |

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✔ All equations solved using either factoring or quadratic formula as appropriate.

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