Certainly! Let's solve the quadratic equation [tex]\(2x^2 + 16x - 8 = 0\)[/tex] by completing the square.
### Step-by-Step Solution
1. Divide the entire equation by the coefficient of [tex]\(x^2\)[/tex] to simplify:
The given equation is:
[tex]\[
2x^2 + 16x - 8 = 0
\][/tex]
Divide every term by 2:
[tex]\[
x^2 + 8x - 4 = 0
\][/tex]
2. Move the constant term to the right side of the equation:
[tex]\[
x^2 + 8x = 4
\][/tex]
3. Complete the square:
To complete the square on the left-hand side, take half of the coefficient of [tex]\(x\)[/tex] (which is 8), square it, and add it to both sides of the equation.
[tex]\[
\left(\frac{8}{2}\right)^2 = 16
\][/tex]
Add 16 to both sides:
[tex]\[
x^2 + 8x + 16 = 4 + 16
\][/tex]
This simplifies to:
[tex]\[
(x + 4)^2 = 20
\][/tex]
4. Take the square root of both sides:
[tex]\[
x + 4 = \pm \sqrt{20}
\][/tex]
Simplify [tex]\(\sqrt{20}\)[/tex]:
[tex]\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
\][/tex]
So:
[tex]\[
x + 4 = \pm 2\sqrt{5}
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4 \pm 2\sqrt{5}
\][/tex]
Therefore, the solutions to the equation [tex]\(2x^2 + 16x - 8 = 0\)[/tex] are:
[tex]\[
x = -4 + 2\sqrt{5} \quad \text{and} \quad x = -4 - 2\sqrt{5}
\][/tex]
### Correct Answer:
The correct answer is:
C. [tex]\(x = -4 \pm 2 \sqrt{5}\)[/tex]
