Question

Fill in the steps to complete the square: x2 +12x − 14 = 0 x2 + 12x = 14 x2 +12x + Response area = 14+ Response area We factor and simplify to get: (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: Response area

Answer 1

Let's complete the square step by step for the equation \( x^2 + 12x - 14 = 0 \).

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

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

3. **Identify the coefficient of \( x \), divide by 2, and square it:**
- Coefficient of \( x \) is 12.
- Half of 12 is 6, and \( 6^2 = 36 \).

4. **Add the squared term to both sides:**
\[
x^2 + 12x + 36 = 14 + 36
\]

5. **Simplify the right side:**
\[
x^2 + 12x + 36 = 50
\]

6. **Factor the left side:**
\[
(x + 6)^2 = 50
\]

7. **Take the square root of both sides:**
- Remember to consider both the positive and negative roots.
\[
x + 6 = \pm \sqrt{50}
\]

8. **Isolate the variable \( x \):**
\[
x = -6 \pm \sqrt{50}
\]

9. **Final answers:**
- Since \( \sqrt{50} = 5\sqrt{2} \), we can rewrite the answers as:
\[
x = -6 + 5\sqrt{2} \quad \text{or} \quad x = -6 - 5\sqrt{2}
\]

So, filled in response areas would be:

- **Response area 1:** \( 36 \)
- **Response area 2:** \( 36 \)
- **Response area 3:** \( \pm \sqrt{50} \)
- **Response area 4:** \( -6 \pm \sqrt{50} \)
- **Response area 5:** \( -6 \pm 5\sqrt{2} \)