Question

Complete the square to solve the following quadratic equation: x^2-6x-16=0
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Answer 1

To solve the quadratic equation \( x^2 - 6x - 16 = 0 \) by completing the square, follow these steps:

1. **Move the constant term to the other side of the equation**:
\[
x^2 - 6x = 16
\]

2. **Complete the square**: Take the coefficient of \( x \) (which is -6), divide it by 2, and square it:
\[
\left(-\frac{6}{2}\right)^2 = (-3)^2 = 9
\]
Now add this square to both sides of the equation:
\[
x^2 - 6x + 9 = 16 + 9
\]
Simplifying both sides, we get:
\[
x^2 - 6x + 9 = 25
\]

3. **Rewrite the left side as a squared binomial**:
\[
(x - 3)^2 = 25
\]

4. **Take the square root of both sides**:
\[
x - 3 = \pm 5
\]

5. **Solve for \( x \)** by isolating \( x \):
- For the positive case:
\[
x - 3 = 5 \implies x = 5 + 3 = 8
\]
- For the negative case:
\[
x - 3 = -5 \implies x = -5 + 3 = -2
\]

Thus, the solutions to the equation \( x^2 - 6x - 16 = 0 \) are:
\[
x = 8 \quad \text{and} \quad x = -2
\]