Select the correct answer.

Solve the equation using the method of completing the square.

[tex]\[ 2x^2 + 16x - 8 = 0 \][/tex]

A. [tex]\[ x = -2 \pm 4 \sqrt{5} \][/tex]

B. [tex]\[ x = 2 \pm 4 \sqrt{5} \][/tex]

C. [tex]\[ x = 4 \pm 2 \sqrt{5} \][/tex]

D. [tex]\[ x = -4 \pm 2 \sqrt{5} \][/tex]



Answer :

To solve the equation [tex]\( 2x^2 + 16x - 8 = 0 \)[/tex] using the method of completing the square, follow these steps:

1. Divide the whole equation by 2 to simplify it:
[tex]\[ \frac{2x^2 + 16x - 8}{2} = 0 \implies x^2 + 8x - 4 = 0 \][/tex]

2. Rearrange the equation to set it in the form where the linear and quadratic terms are grouped together:
[tex]\[ x^2 + 8x = 4 \][/tex]

3. Complete the square by adding and subtracting the square of half the coefficient of [tex]\( x \)[/tex]. Here, the coefficient of [tex]\( x \)[/tex] is 8, so [tex]\(\left( \frac{8}{2} \right)^2 = 16\)[/tex]:
[tex]\[ x^2 + 8x + 16 - 16 = 4 \][/tex]
Which simplifies to:
[tex]\[ (x + 4)^2 - 16 = 4 \][/tex]

4. Isolate the perfect square term by adding 16 to both sides:
[tex]\[ (x + 4)^2 = 20 \][/tex]

5. Solve for [tex]\( x \)[/tex] by taking the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[ x + 4 = \pm \sqrt{20} \][/tex]
Since [tex]\( \sqrt{20} \)[/tex] can be simplified to [tex]\( 2\sqrt{5} \)[/tex]:
[tex]\[ x + 4 = \pm 2\sqrt{5} \][/tex]

6. Solve for [tex]\( x \)[/tex] by subtracting 4 from both sides:
[tex]\[ x = -4 \pm 2\sqrt{5} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{x = -4 \pm 2\sqrt{5}} \][/tex]

So, the correct option is:
[tex]\[ \boxed{D. \, x = -4 \pm 2\sqrt{5}} \][/tex]