Question

Solve the equation by completing the square.xsquaredplus4xequals32Question content area bottomPart 1Select the correct choice and fill in any answer boxes in your choice below.

Answer 1

To solve the equation \( x^2 + 4x = 32 \) by completing the square, follow these steps:

1. **Move the constant to the other side of the equation:**
\[
x^2 + 4x - 32 = 0
\]

2. **Isolate the \( x \) terms:**
\[
x^2 + 4x = 32
\]

3. **Complete the square:**
To complete the square for the expression \( x^2 + 4x \), take half of the coefficient of \( x \) (which is \( 4 \)), square it, and add it to both sides.

Half of \( 4 \) is \( 2 \), and squaring it gives \( 4 \).

Add \( 4 \) to both sides:
\[
x^2 + 4x + 4 = 32 + 4
\]
\[
x^2 + 4x + 4 = 36
\]

4. **Rewrite the left side as a square:**
The left side \( x^2 + 4x + 4 \) can be written as \( (x + 2)^2 \):
\[
(x + 2)^2 = 36
\]

5. **Take the square root of both sides:**
\[
x + 2 = \pm 6
\]

6. **Solve for \( x \):**
This gives us two cases:
\[
x + 2 = 6 \quad \Rightarrow \quad x = 4
\]
\[
x + 2 = -6 \quad \Rightarrow \quad x = -8
\]

Therefore, the solutions to the equation \( x^2 + 4x = 32 \) are:
\[
x = 4 \quad \text{and} \quad x = -8
\]