Question

Complete the square to identify the two values that solve the following quadratic equation: x2−16x−13=4.(1 point)

Answer 1

To solve the quadratic equation \(x^2 - 16x - 13 = 4\) by completing the square, we first rearrange the equation:

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

This simplifies to:

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

Next, we will complete the square for the expression \(x^2 - 16x\). To do this, we take the coefficient of \(x\), which is \(-16\), divide it by 2 to get \(-8\), and then square it to get \(64\).

Now, we add and subtract \(64\) inside the equation:

\[
x^2 - 16x + 64 - 64 - 17 = 0
\]

This simplifies to:

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

Next, we isolate the square:

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

Now, we'll take the square root of both sides:

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

Solving for \(x\) in each case gives:

1. \(x - 8 = 9\) leads to \(x = 17\).
2. \(x - 8 = -9\) leads to \(x = -1\).

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

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