Completing The Square Worksheet Unique Pleting The Square ... - Free Printable

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:
\[ 24 = \frac{96}{4} \]
\[ x^2 - 5x + \frac{25}{4} = \frac{96}{4} + \frac{25}{4} \]
\[ x^2 - 5x + \frac{25}{4} = \frac{121}{4} \]

#### 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:

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