Question

Complete the square to identify the two values that solve the following quadratic equation x squared - 16x - 13 = 4

Answer 1

To solve the equation \( x^2 - 16x - 13 = 4 \) by completing the square, we first need to rearrange the equation so that it equals 0:

\[
x^2 - 16x - 13 - 4 = 0
\]
\[
x^2 - 16x - 17 = 0
\]

Next, we complete the square for the quadratic expression \( x^2 - 16x \).

1. Take the coefficient of \( x \), which is \(-16\), divide by 2 to get \(-8\), and then square it:

\[
\left(-8\right)^2 = 64
\]

2. We rewrite the expression \( x^2 - 16x \) as follows:

\[
x^2 - 16x = (x - 8)^2 - 64
\]

3. Substitute this back into the equation:

\[
(x - 8)^2 - 64 - 17 = 0
\]
\[
(x - 8)^2 - 81 = 0
\]

4. Now isolate the perfect square:

\[
(x - 8)^2 = 81
\]

5. Take the square root of both sides:

\[
x - 8 = 9 \quad \text{or} \quad x - 8 = -9
\]

6. Solving for \( x \):

\[
x = 9 + 8 = 17 \quad \text{or} \quad x = -9 + 8 = -1
\]

Therefore, the two values that solve the equation are:

\[
x = 17 \quad \text{and} \quad x = -1
\]