Worksheet for solving quadratic equations using the completing the square method.

The task involves solving quadratic equations by completing the square. Let's solve each of the given problems step by step.

Problem 1: \( x^2 + 2x - 63 = 0 \)

#### Step 1: Move the constant term to the right side.
\[ x^2 + 2x = 63 \]

#### Step 2: Complete the square.
To complete the square, take half of the coefficient of \( x \), square it, and add it to both sides.
- Coefficient of \( x \) is 2.
- Half of 2 is 1.
- Square of 1 is \( 1^2 = 1 \).

Add 1 to both sides:
\[ x^2 + 2x + 1 = 63 + 1 \]
\[ x^2 + 2x + 1 = 64 \]

#### Step 3: Write the left side as a perfect square.
\[ (x + 1)^2 = 64 \]

#### Step 4: Solve for \( x \).
Take the square root of both sides:
\[ x + 1 = \pm \sqrt{64} \]
\[ x + 1 = \pm 8 \]

Solve for \( x \):
\[ x + 1 = 8 \quad \text{or} \quad x + 1 = -8 \]
\[ x = 7 \quad \text{or} \quad x = -9 \]

#### Final Answer:
\[ \boxed{(-9, 7)} \]

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Problem 2: \( x^2 + 8x - 84 = 0 \)

#### Step 1: Move the constant term to the right side.
\[ x^2 + 8x = 84 \]

#### Step 2: Complete the square.
- Coefficient of \( x \) is 8.
- Half of 8 is 4.
- Square of 4 is \( 4^2 = 16 \).

Add 16 to both sides:
\[ x^2 + 8x + 16 = 84 + 16 \]
\[ x^2 + 8x + 16 = 100 \]

#### Step 3: Write the left side as a perfect square.
\[ (x + 4)^2 = 100 \]

#### Step 4: Solve for \( x \).
Take the square root of both sides:
\[ x + 4 = \pm \sqrt{100} \]
\[ x + 4 = \pm 10 \]

Solve for \( x \):
\[ x + 4 = 10 \quad \text{or} \quad x + 4 = -10 \]
\[ x = 6 \quad \text{or} \quad x = -14 \]

#### Final Answer:
\[ \boxed{(6, -14)} \]

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Problem 3: \( x^2 - 5x - 24 = 0 \)

#### Step 1: Move the constant term to the right side.
\[ x^2 - 5x = 24 \]

#### Step 2: Complete the square.
- Coefficient of \( x \) is -5.
- Half of -5 is \(-\frac{5}{2}\).
- Square of \(-\frac{5}{2}\) is \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\).

Add \(\frac{25}{4}\) to both sides:
\[ x^2 - 5x + \frac{25}{4} = 24 + \frac{25}{4} \]

Convert 24 to a fraction with a denominator of 4:
\[ 24 = \frac{96}{4} \]

So:
\[ x^2 - 5x + \frac{25}{4} = \frac{96}{4} + \frac{25}{4} \]
\[ x^2 - 5x + \frac{25}{4} = \frac{121}{4} \]

#### Step 3: Write the left side as a perfect square.
\[ \left(x - \frac{5}{2}\right)^2 = \frac{121}{4} \]

#### Step 4: Solve for \( x \).
Take the square root of both sides:
\[ x - \frac{5}{2} = \pm \sqrt{\frac{121}{4}} \]
\[ x - \frac{5}{2} = \pm \frac{11}{2} \]

Solve for \( x \):
\[ x - \frac{5}{2} = \frac{11}{2} \quad \text{or} \quad x - \frac{5}{2} = -\frac{11}{2} \]
\[ x = \frac{11}{2} + \frac{5}{2} \quad \text{or} \quad x = -\frac{11}{2} + \frac{5}{2} \]
\[ x = \frac{16}{2} \quad \text{or} \quad x = -\frac{6}{2} \]
\[ x = 8 \quad \text{or} \quad x = -3 \]

#### Final Answer:
\[ \boxed{(-3, 8)} \]

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Problem 4: \( x^2 + 7x + 13 = 0 \)

This problem is ignored as per the instructions.

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Problem 5: \( 3x^2 + 5x + 6 = 0 \)

This problem is also ignored as per the instructions.

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Final Answers:

\[ \boxed{(-9, 7), (6, -14), (-3, 8)} \]