Question

Complete the square to identify the two values that solve the following quadratic equation: x^2−16x−13=4
why is the correct answer 17, -1
.

Answer 1

To solve the quadratic equation \( x^2 - 16x - 13 = 4 \) by completing the square, follow these steps:

1. **Rearrange the equation**:

First, we want to set the equation to zero:
\[
x^2 - 16x - 13 - 4 = 0 \quad \Rightarrow \quad x^2 - 16x - 17 = 0
\]

2. **Move the constant term to the other side**:

We want to complete the square for the \(x\) terms:
\[
x^2 - 16x = 17
\]

3. **Complete the square**:

To complete the square, take half of the coefficient of \(x\) (which is \(-16\)), square it, and then add it to both sides. Half of \(-16\) is \(-8\), and squaring it gives:
\[
(-8)^2 = 64
\]

Add \(64\) to both sides:
\[
x^2 - 16x + 64 = 17 + 64
\]
\[
x^2 - 16x + 64 = 81
\]

4. **Factor the left side**:

The left side factors to:
\[
(x - 8)^2 = 81
\]

5. **Take the square root of both sides**:

Now, take the square root of both sides:
\[
x - 8 = \pm 9
\]

This gives two equations:
\[
x - 8 = 9 \quad \Rightarrow \quad x = 17
\]
\[
x - 8 = -9 \quad \Rightarrow \quad x = -1
\]

6. **The solutions**:

Thus, the two solutions to the equation \(x^2 - 16x - 13 = 4\) are:
\[
\boxed{17} \quad \text{and} \quad \boxed{-1}
\]

So the answer of 17 and -1 is correct.