Math worksheet for solving quadratic equations using the completing the square method.

We are given 10 quadratic equations to solve by completing the square. Let’s go through each one step-by-step.

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General Method: Completing the Square

For a quadratic equation in the form:

> x² + bx = c

1. Take half of the coefficient of x → (b/2)
2. Square it → (b/2)²
3. Add this square to both sides of the equation.
4. The left side becomes a perfect square trinomial: (x + b/2)²
5. Take the square root of both sides.
6. Solve for x.

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## Problem 1: x² + 6x = 16

- Half of 6 is 3 → 3² = 9
- Add 9 to both sides:

> x² + 6x + 9 = 16 + 9
> (x + 3)² = 25

- Take square roots:

> x + 3 = ±5

- Solve:

> x = -3 ± 5 → x = 2 or x = -8

✔ Answer: x = 2, -8

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## Problem 2: x² – 10x – 39 = 0

First, move constant to the other side:

> x² – 10x = 39

- Half of -10 is -5 → (-5)² = 25
- Add 25 to both sides:

> x² – 10x + 25 = 39 + 25
> (x – 5)² = 64

- Square root:

> x – 5 = ±8

- Solve:

> x = 5 ± 8 → x = 13 or x = -3

✔ Answer: x = 13, -3

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## Problem 3: x² – 6x = 16

- Half of -6 is -3 → (-3)² = 9
- Add 9 to both sides:

> x² – 6x + 9 = 16 + 9
> (x – 3)² = 25

- Square root:

> x – 3 = ±5

- Solve:

> x = 3 ± 5 → x = 8 or x = -2

✔ Answer: x = 8, -2

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## Problem 4: x² + 4x = 0

- Half of 4 is 2 → 2² = 4
- Add 4 to both sides:

> x² + 4x + 4 = 0 + 4
> (x + 2)² = 4

- Square root:

> x + 2 = ±2

- Solve:

> x = -2 ± 2 → x = 0 or x = -4

✔ Answer: x = 0, -4

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## Problem 5: x² – 8x = -10

- Half of -8 is -4 → (-4)² = 16
- Add 16 to both sides:

> x² – 8x + 16 = -10 + 16
> (x – 4)² = 6

- Square root:

> x – 4 = ±√6

- Solve:

> x = 4 ± √6

✔ Answer: x = 4 + √6, 4 – √6

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## Problem 6: x² + 2x = 3

- Half of 2 is 1 → 1² = 1
- Add 1 to both sides:

> x² + 2x + 1 = 3 + 1
> (x + 1)² = 4

- Square root:

> x + 1 = ±2

- Solve:

> x = -1 ± 2 → x = 1 or x = -3

✔ Answer: x = 1, -3

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## Problem 7: x² + 6x + 8 = 0

Move constant to other side:

> x² + 6x = -8

- Half of 6 is 3 → 3² = 9
- Add 9 to both sides:

> x² + 6x + 9 = -8 + 9
> (x + 3)² = 1

- Square root:

> x + 3 = ±1

- Solve:

> x = -3 ± 1 → x = -2 or x = -4

✔ Answer: x = -2, -4

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## Problem 8: x² + 10x = 55

- Half of 10 is 5 → 5² = 25
- Add 25 to both sides:

> x² + 10x + 25 = 55 + 25
> (x + 5)² = 80

- Square root:

> x + 5 = ±√80 = ±√(16×5) = ±4√5

- Solve:

> x = -5 ± 4√5

✔ Answer: x = -5 + 4√5, -5 – 4√5

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## Problem 9: x² – 2x – 3 = 0

Move constant:

> x² – 2x = 3

- Half of -2 is -1 → (-1)² = 1
- Add 1 to both sides:

> x² – 2x + 1 = 3 + 1
> (x – 1)² = 4

- Square root:

> x – 1 = ±2

- Solve:

> x = 1 ± 2 → x = 3 or x = -1

✔ Answer: x = 3, -1

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## Problem 10: x² = 12x

Bring all terms to one side:

> x² – 12x = 0

- Half of -12 is -6 → (-6)² = 36
- Add 36 to both sides:

> x² – 12x + 36 = 0 + 36
> (x – 6)² = 36

- Square root:

> x – 6 = ±6

- Solve:

> x = 6 ± 6 → x = 12 or x = 0

✔ Answer: x = 12, 0

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## ✔ Final Answers:

1. x = 2, -8
2. x = 13, -3
3. x = 8, -2
4. x = 0, -4
5. x = 4 ± √6
6. x = 1, -3
7. x = -2, -4
8. x = -5 ± 4√5
9. x = 3, -1
10. x = 12, 0

Let me know if you’d like to see any step explained further!