To determine which of the given options are solutions to the quadratic equation [tex]\( x^2 + 2x - 8 = 0 \)[/tex], we'll need to solve the equation and then check each option against the solutions.
### Step-by-Step Solution:
1. Write down the quadratic equation:
[tex]\[
x^2 + 2x - 8 = 0
\][/tex]
2. Solve the quadratic equation:
The solutions to a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In this equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -8 \)[/tex].
3. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = 2^2 - 4(1)(-8) = 4 + 32 = 36
\][/tex]
4. Plug the discriminant back into the quadratic formula:
[tex]\[
x = \frac{-2 \pm \sqrt{36}}{2 \cdot 1} = \frac{-2 \pm 6}{2}
\][/tex]
5. Find the two solutions:
[tex]\[
x = \frac{-2 + 6}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
x = \frac{-2 - 6}{2} = \frac{-8}{2} = -4
\][/tex]
Thus, the solutions to the quadratic equation [tex]\( x^2 + 2x - 8 = 0 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = -4 \)[/tex].
### Check the Given Options:
Let's check each of the given options to see if they match the solutions [tex]\( x = 2 \)[/tex] and [tex]\( x = -4 \)[/tex]:
- A. [tex]\(-1\)[/tex]: [tex]\(-1\)[/tex] is not in the list of solutions.
- B. [tex]\(6\)[/tex]: [tex]\(6\)[/tex] is not in the list of solutions.
- C. [tex]\(-4\)[/tex]: [tex]\(-4\)[/tex] is a solution.
- D. [tex]\(2\)[/tex]: [tex]\(2\)[/tex] is a solution.
- E. [tex]\(8\)[/tex]: [tex]\(8\)[/tex] is not in the list of solutions.
### Summary:
The options that are solutions to the quadratic equation [tex]\( x^2 + 2x - 8 = 0 \)[/tex] are:
- [tex]\( C. -4 \)[/tex]
- [tex]\( D. 2 \)[/tex]
So, the correct options are [tex]\( \boxed{C} \)[/tex] and [tex]\( \boxed{D} \)[/tex].
