Question

Complete the square to identify the two values that solve the following quadratic equation: x2−16x−13=4
.

Answer 1

To solve the quadratic equation \( x^2 - 16x - 13 = 4 \) using the method of completing the square, we first set the equation in the standard form:

1. **Rearrange the equation**:
\[
x^2 - 16x - 13 - 4 = 0
\]
\[
x^2 - 16x - 17 = 0
\]

2. **Move the constant to the other side**:
\[
x^2 - 16x = 17
\]

3. **Complete the square**:
- Take the coefficient of \( x \), which is \(-16\).
- Divide it by 2: \(-16 \div 2 = -8\).
- Square the result: \((-8)^2 = 64\).

4. **Add and subtract this square on the left side of the equation**:
\[
x^2 - 16x + 64 = 17 + 64
\]
\[
(x - 8)^2 = 81
\]

5. **Take the square root of both sides**:
\[
x - 8 = \pm 9
\]

6. **Solve for \( x \)**:
- For the positive case:
\[
x - 8 = 9 \implies x = 9 + 8 = 17
\]
- For the negative case:
\[
x - 8 = -9 \implies x = -9 + 8 = -1
\]

Thus, the two values that solve the equation \( x^2 - 16x - 13 = 4 \) are:

\[
\boxed{17 \text{ and } -1}
\]