Completing the square worksheet: Fill out & sign online | DocHub - Free Printable

Let's solve each problem step-by-step using the method of completing the square. We'll also address the last two questions about creating perfect square trinomials.

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🔷 Completing the Square Method Overview:

To complete the square for a quadratic equation in the form:

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

1. Move constant to the other side:
$$
x^2 + bx = -c
$$

2. Take half of the coefficient of $ x $, square it, and add to both sides:
$$
\left(\frac{b}{2}\right)^2
$$

3. Factor the left side into a perfect square trinomial.

4. Solve by taking square roots.

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Now, let’s go through each problem.

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1. $ x^2 + 8x - 6 = 0 $

Step 1: Move constant:
$$
x^2 + 8x = 6
$$

Step 2: Half of 8 is 4; square it: $ 4^2 = 16 $. Add to both sides:
$$
x^2 + 8x + 16 = 6 + 16 = 22
$$

Step 3: Factor:
$$
(x + 4)^2 = 22
$$

Step 4: Solve:
$$
x + 4 = \pm\sqrt{22} \Rightarrow x = -4 \pm \sqrt{22}
$$

✔ Answer: $ x = -4 \pm \sqrt{22} $

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2. $ x^2 + 6x + 9 = 0 $

This one is already a perfect square!

$$
x^2 + 6x + 9 = (x + 3)^2 = 0
$$

So,
$$
x + 3 = 0 \Rightarrow x = -3
$$

✔ Answer: $ x = -3 $

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3. $ 2x^2 - 20x - 64 = 0 $

First, divide entire equation by 2 to make leading coefficient 1:
$$
x^2 - 10x - 32 = 0
$$

Move constant:
$$
x^2 - 10x = 32
$$

Half of -10 is -5; square it: $ (-5)^2 = 25 $. Add to both sides:
$$
x^2 - 10x + 25 = 32 + 25 = 57
$$

Factor:
$$
(x - 5)^2 = 57
$$

Solve:
$$
x - 5 = \pm\sqrt{57} \Rightarrow x = 5 \pm \sqrt{57}
$$

✔ Answer: $ x = 5 \pm \sqrt{57} $

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4. $ x^2 + 16x + 24 = 0 $

Move constant:
$$
x^2 + 16x = -24
$$

Half of 16 is 8; $ 8^2 = 64 $. Add to both sides:
$$
x^2 + 16x + 64 = -24 + 64 = 40
$$

Factor:
$$
(x + 8)^2 = 40
$$

Solve:
$$
x + 8 = \pm\sqrt{40} = \pm 2\sqrt{10} \Rightarrow x = -8 \pm 2\sqrt{10}
$$

✔ Answer: $ x = -8 \pm 2\sqrt{10} $

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5. $ x^2 + 4x + 10 = 0 $

Move constant:
$$
x^2 + 4x = -10
$$

Half of 4 is 2; $ 2^2 = 4 $. Add to both sides:
$$
x^2 + 4x + 4 = -10 + 4 = -6
$$

Factor:
$$
(x + 2)^2 = -6
$$

Solve:
$$
x + 2 = \pm\sqrt{-6} = \pm i\sqrt{6} \Rightarrow x = -2 \pm i\sqrt{6}
$$

✔ Answer: $ x = -2 \pm i\sqrt{6} $ (complex solutions)

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6. $ x^2 + 8x + 7 = 0 $

Move constant:
$$
x^2 + 8x = -7
$$

Half of 8 is 4; $ 4^2 = 16 $. Add to both sides:
$$
x^2 + 8x + 16 = -7 + 16 = 9
$$

Factor:
$$
(x + 4)^2 = 9
$$

Solve:
$$
x + 4 = \pm 3 \Rightarrow x = -4 \pm 3
$$

So:
- $ x = -4 + 3 = -1 $
- $ x = -4 - 3 = -7 $

✔ Answer: $ x = -1 $ or $ x = -7 $

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7. $ x^2 + 24x - 16 = 0 $

Move constant:
$$
x^2 + 24x = 16
$$

Half of 24 is 12; $ 12^2 = 144 $. Add to both sides:
$$
x^2 + 24x + 144 = 16 + 144 = 160
$$

Factor:
$$
(x + 12)^2 = 160
$$

Solve:
$$
x + 12 = \pm\sqrt{160} = \pm\sqrt{16 \cdot 10} = \pm 4\sqrt{10}
\Rightarrow x = -12 \pm 4\sqrt{10}
$$

✔ Answer: $ x = -12 \pm 4\sqrt{10} $

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8. $ x^2 + 4x - 3 = 0 $

Move constant:
$$
x^2 + 4x = 3
$$

Half of 4 is 2; $ 2^2 = 4 $. Add to both sides:
$$
x^2 + 4x + 4 = 3 + 4 = 7
$$

Factor:
$$
(x + 2)^2 = 7
$$

Solve:
$$
x + 2 = \pm\sqrt{7} \Rightarrow x = -2 \pm \sqrt{7}
$$

✔ Answer: $ x = -2 \pm \sqrt{7} $

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9. What value completes $ x^2 + 16x + \boxed{\phantom{0}} $ to make a perfect square trinomial?

We use the formula: take half of the middle term, square it.

- Coefficient of $ x $ is 16.
- Half: $ \frac{16}{2} = 8 $
- Square: $ 8^2 = 64 $

So, $ x^2 + 16x + 64 = (x + 8)^2 $

✔ Answer: $ \boxed{64} $

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10. What value completes $ x^2 - 3x + \boxed{\phantom{0}} $?

- Coefficient of $ x $ is -3.
- Half: $ \frac{-3}{2} $
- Square: $ \left(\frac{-3}{2}\right)^2 = \frac{9}{4} $

So, $ x^2 - 3x + \frac{9}{4} = \left(x - \frac{3}{2}\right)^2 $

✔ Answer: $ \boxed{\frac{9}{4}} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ x = -4 \pm \sqrt{22} $ |
| 2 | $ x = -3 $ |
| 3 | $ x = 5 \pm \sqrt{57} $ |
| 4 | $ x = -8 \pm 2\sqrt{10} $ |
| 5 | $ x = -2 \pm i\sqrt{6} $ |
| 6 | $ x = -1 $ or $ x = -7 $ |
| 7 | $ x = -12 \pm 4\sqrt{10} $ |
| 8 | $ x = -2 \pm \sqrt{7} $ |
| 9 | $ \boxed{64} $ |
| 10 | $ \boxed{\frac{9}{4}} $ |

Let me know if you'd like these written neatly on a printable format!