Question

Solve the equation for all values of x by completing the square.
4, x, squared, plus, 32, x, plus, 24, equals, 0
4x
2
+32x+24=
0

Answer 1

To solve the equation \( 4x^2 + 32x + 24 = 0 \) by completing the square, follow these steps:

1. **Divide the equation by 4** to simplify it:

\[
x^2 + 8x + 6 = 0
\]

2. **Move the constant term to the other side**:

\[
x^2 + 8x = -6
\]

3. **Complete the square**. Take half of the coefficient of \(x\) (which is 8), square it, and add to both sides:

Half of 8 is 4, and \(4^2 = 16\).

\[
x^2 + 8x + 16 = -6 + 16
\]

Simplifying the right side:

\[
x^2 + 8x + 16 = 10
\]

4. **Rewrite the left-hand side as a square**:

\[
(x + 4)^2 = 10
\]

5. **Take the square root of both sides**:

\[
x + 4 = \pm\sqrt{10}
\]

6. **Solve for \(x\)**:

\[
x = -4 + \sqrt{10} \quad \text{or} \quad x = -4 - \sqrt{10}
\]

Thus, the solutions to the equation \( 4x^2 + 32x + 24 = 0 \) are:

\[
x = -4 + \sqrt{10}
\]
\[
x = -4 - \sqrt{10}
\]