Question

Solve the equation for all values of x by completing the square. Express your answer in simplest form.
x, squared, plus, 12, x, plus, 26, equals, 0
x
2
+12x+26=
0

Answer 1

To solve the equation \( x^2 + 12x + 26 = 0 \) by completing the square, follow these steps:

1. **Rearrange the terms:**
\[
x^2 + 12x = -26
\]

2. **Complete the square:**
- Take the coefficient of \( x \) (which is 12), divide it by 2 to get 6, and then square it to get \( 6^2 = 36 \).
- Add 36 to both sides of the equation:
\[
x^2 + 12x + 36 = -26 + 36
\]
Simplifying the right side:
\[
x^2 + 12x + 36 = 10
\]

3. **Rewrite the left side as a square:**
\[
(x + 6)^2 = 10
\]

4. **Take the square root of both sides:**
\[
x + 6 = \pm \sqrt{10}
\]

5. **Isolate \( x \):**
- For the positive case:
\[
x = -6 + \sqrt{10}
\]
- For the negative case:
\[
x = -6 - \sqrt{10}
\]

6. **Final solutions:**
\[
x = -6 + \sqrt{10} \quad \text{and} \quad x = -6 - \sqrt{10}
\]

Therefore, the solutions to the equation \( x^2 + 12x + 26 = 0 \) are:
\[
x = -6 + \sqrt{10} \quad \text{and} \quad x = -6 - \sqrt{10}
\]