Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, plus, 16, x, equals, minus, 51
x
2
+16x=
−51

Answer 1

To solve the equation \( x^2 + 16x = -51 \) by completing the square, we follow these steps:

1. Move the constant term to the right side of the equation:

\[
x^2 + 16x + 51 = 0
\]

2. To complete the square on the left side, we take the coefficient of \( x \) (which is \( 16 \)), divide it by \( 2 \) (getting \( 8 \)), and then square it (getting \( 64 \)).

3. Add and subtract \( 64 \) in the equation:

\[
x^2 + 16x + 64 - 64 + 51 = 0
\]

This simplifies to:

\[
(x + 8)^2 - 13 = 0
\]

4. Now isolate the squared term:

\[
(x + 8)^2 = 13
\]

5. Take the square root of both sides:

\[
x + 8 = \pm \sqrt{13}
\]

6. Finally, solve for \( x \):

\[
x = -8 \pm \sqrt{13}
\]

Thus, the solutions are:

\[
x = -8 + \sqrt{13} \quad \text{and} \quad x = -8 - \sqrt{13}
\]

In simplified form, the solutions are:

\[
x = -8 + \sqrt{13}, \quad x = -8 - \sqrt{13}
\]