To solve for [tex]\(x\)[/tex] in the quadratic equation using the quadratic formula, we need to evaluate the expression:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Given [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = -8\)[/tex], the first step is to calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = (-2)^2 - 4(1)(-8)
\][/tex]
Calculating inside the discriminant:
[tex]\[
\text{Discriminant} = 4 + 32 = 36
\][/tex]
Next, we find the square root of the discriminant:
[tex]\[
\sqrt{36} = 6
\][/tex]
Now, we substitute back into the quadratic formula:
[tex]\[
x = \frac{-(-2) \pm \sqrt{36}}{2(1)}
\][/tex]
Simplify the expressions:
[tex]\[
x = \frac{2 \pm 6}{2}
\][/tex]
Therefore, the number needed for our problem is 6, because:
[tex]\[
x = \frac{2 \pm [?]}{[]}
\][/tex]
We find that [tex]\( \pm ? = 6 \)[/tex].
So, the number that belongs in the green box is:
[tex]\[
\boxed{6}
\][/tex]
