Question

Fill in the steps to complete the square: x2 +12x − 14 = 0x2 +12x − 14 = 0 x2 + 12x = 14x2 + 12x = 14 x2 +12x + x2 +12x + Response area = 14+ = 14+ Response area We factor and simplify to get: (x+6)2 =(x+6)2 = Response area At this point, we need to Response area After that is done, we will have: Response area == Response area We get isolate the variable by: Response area Final answers are:

Answer 1

To complete the square for the equation \(x^2 + 12x - 14 = 0\) step-by-step, let's fill in each of the gaps:

1. Start with the equation:
\[
x^2 + 12x - 14 = 0
\]

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

3. To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is 12. Half of 12 is 6, and squaring it gives 36:
\[
x^2 + 12x + 36 = 14 + 36
\]

4. Simplify both sides:
\[
x^2 + 12x + 36 = 50
\]

5. We can now factor the left side:
\[
(x + 6)^2 = 50
\]

6. At this point, we need to take the square root of both sides:
\[
x + 6 = \pm \sqrt{50}
\]

7. Simplifying \( \sqrt{50} \) gives:
\[
\sqrt{50} = 5\sqrt{2}
\]
So, we write:
\[
x + 6 = \pm 5\sqrt{2}
\]

8. Now isolate the variable \(x\):
\[
x = -6 \pm 5\sqrt{2}
\]

9. Final answers are:
\[
x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2}
\]

In summary, the steps have filled in as follows:

- Starting equation: \(x^2 + 12x - 14 = 0\)
- Rearranged to: \(x^2 + 12x = 14\)
- Completing the square: \(x^2 + 12x + 36 = 50\)
- Factoring gives: \((x + 6)^2 = 50\)
- Taking the square root: \(x + 6 = \pm 5\sqrt{2}\)
- Isolate \(x\): \(x = -6 \pm 5\sqrt{2}\)

The final answers are thus:
\[
x = -6 + 5\sqrt{2}, \quad x = -6 - 5\sqrt{2}
\]