Question

Fill in the steps to complete the square x ^2 + 12x - 14 = 0
x^2 + 12x = 14
x^2 + 12x + blank = 14 + blank

50
13 Take the square root of both sides of the equation
Square both sides of the equation
100
(x + 6)^2
Square root x + 6
Subtracting twelve from both sides of the equation
square root x + 12
-6 greater than or equal to square root 36
Subtracting seats from both sides of the equation
- 6 greater than or equal to square root 50
144
Greater than or equal to 12
Greater than or equal to square root 36
(x + 12) ^2
x + 6
Greater than or equal to square root 50
36
x + 12
17
122
-22
-12&0

Answer 1

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

1. **Isolate the quadratic and linear terms**:
\[
x^2 + 12x = 14
\]

2. **Complete the square**:
To complete the square, we need to add and subtract \(\left(\frac{12}{2}\right)^2\) (which is \( 36 \)) to the left side of the equation.

Therefore, we will add \( 36 \) to both sides:
\[
x^2 + 12x + 36 = 14 + 36
\]

3. **Rewrite the left side as a square**:
The left side can now be written as a perfect square:
\[
(x + 6)^2 = 50
\]

4. **Take the square root of both sides**:
\[
x + 6 = \pm \sqrt{50}
\]

Since \(\sqrt{50} = 5\sqrt{2}\), we can express it more simply:
\[
x + 6 = \pm 5\sqrt{2}
\]

5. **Solve for \(x\)**:
Subtract \(6\) from both sides:
\[
x = -6 \pm 5\sqrt{2}
\]

So, the complete solution to the original equation is:
\[
x = -6 + 5\sqrt{2} \quad \text{or} \quad x = -6 - 5\sqrt{2}
\]

As we described the steps, the correct entries for what's missing would be:

- In the completing the square step (i.e., the blanks): "36" for both blanks.
- The square root result before solving would have necessitated \( \sqrt{50} \), which simplifies to \( 5\sqrt{2} \).

Let me know if you need any additional information!