Question

Complete the square: x^2 + 12x - 14 = 0

Answer 1

To complete the square for the equation \( x^2 + 12x - 14 = 0 \), follow these steps:

1. **Move the constant term to the other side**:
\[
x^2 + 12x = 14
\]

2. **Take half of the coefficient of \(x\)**, which is \(12\), divide it by \(2\) to get \(6\), and then square it:
\[
\left(\frac{12}{2}\right)^2 = 6^2 = 36
\]

3. **Add and subtract this square to the left side**:
\[
x^2 + 12x + 36 - 36 = 14
\]
This simplifies to:
\[
(x + 6)^2 - 36 = 14
\]

4. **Now, add \(36\) to both sides** to balance the equation:
\[
(x + 6)^2 = 14 + 36
\]
\[
(x + 6)^2 = 50
\]

5. **Take the square root of both sides**:
\[
x + 6 = \pm \sqrt{50}
\]
Simplifying \(\sqrt{50}\):
\[
\sqrt{50} = 5\sqrt{2}
\]
Thus:
\[
x + 6 = \pm 5\sqrt{2}
\]

6. **Isolate \(x\)**:
\[
x = -6 \pm 5\sqrt{2}
\]

So the solutions to the equation \(x^2 + 12x - 14 = 0\) after completing the square are:
\[
x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2}
\]

Answer 2

can \(-6\pm\sqrt{50}\) be an answer?