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: \[ x = 4 \quad \text{and} \quad x = -8 \]