To solve the quadratic equation [tex]\(2x^2 + 16x - 8 = 0\)[/tex] using the method of completing the square, follow these steps:
### Step 1: Simplify the equation
First, divide the entire equation by 2 to simplify it:
[tex]\[ x^2 + 8x - 4 = 0 \][/tex]
### Step 2: Move the constant term to the other side
Rearrange the equation to move the constant term to the other side:
[tex]\[ x^2 + 8x = 4 \][/tex]
### Step 3: Complete the square
Take half of the coefficient of [tex]\(x\)[/tex] (which is 8), divide it by 2, and then square it. The coefficient of [tex]\(x\)[/tex] is 8, so:
[tex]\[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \][/tex]
Add 16 to both sides of the equation:
[tex]\[ x^2 + 8x + 16 = 4 + 16 \][/tex]
[tex]\[ x^2 + 8x + 16 = 20 \][/tex]
### Step 4: Express as a perfect square
Now, the left side of the equation is a perfect square trinomial:
[tex]\[ (x + 4)^2 = 20 \][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Take the square root of both sides:
[tex]\[ x + 4 = \pm\sqrt{20} \][/tex]
Note that [tex]\(\sqrt{20} = 2\sqrt{5}\)[/tex], so we have:
[tex]\[ x + 4 = \pm2\sqrt{5} \][/tex]
Isolate [tex]\(x\)[/tex] by subtracting 4 from both sides:
[tex]\[ x = -4 \pm 2\sqrt{5} \][/tex]
### Conclusion
The correct solution to the equation is:
[tex]\[ x = -4 \pm 2\sqrt{5} \][/tex]
Therefore, the correct answer is
[tex]\[ \boxed{B. x = -4 \pm 2\sqrt{5}} \][/tex]
