Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, plus, 16, x, plus, 51, equals, 0
x
2
+16x+51=
0
Answer
Attempt 1 out of 5

Answer 1

To solve the equation \( x^2 + 16x + 51 = 0 \) by completing the square, follow these steps:

1. Start with the equation:
\[
x^2 + 16x + 51 = 0
\]

2. Move the constant term to the other side:
\[
x^2 + 16x = -51
\]

3. To complete the square, take half of the coefficient of \( x \) (which is 16), square it, and add it to both sides:
\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\]

4. Add 64 to both sides:
\[
x^2 + 16x + 64 = -51 + 64
\]
\[
x^2 + 16x + 64 = 13
\]

5. The left side can now be factored as a perfect square:
\[
(x + 8)^2 = 13
\]

6. Take the square root of both sides:
\[
x + 8 = \pm \sqrt{13}
\]

7. Solve for \( x \):
\[
x = -8 + \sqrt{13} \quad \text{or} \quad x = -8 - \sqrt{13}
\]

Thus, the solutions to the equation \( x^2 + 16x + 51 = 0 \) are:
\[
x = -8 + \sqrt{13} \quad \text{and} \quad x = -8 - \sqrt{13}
\]