To find the correct substitution of the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula, let's start by rewriting the given equation [tex]\(-2 = -x + x^2 - 4\)[/tex] in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
First, rewrite the equation:
[tex]\[
-2 = -x + x^2 - 4
\][/tex]
Move all terms to the left side of the equation to set it to [tex]\(0\)[/tex]:
[tex]\[
x^2 - x - 4 + 2 = 0
\][/tex]
Simplify it:
[tex]\[
x^2 - x - 2 = 0
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -2\)[/tex].
Now, substitute these values into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -2\)[/tex]:
[tex]\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}
\][/tex]
Simplify it step-by-step:
1. Calculate [tex]\(-b\)[/tex]:
[tex]\[
-(-1) = 1
\][/tex]
2. Calculate the discriminant:
[tex]\[
(-1)^2 - 4(1)(-2) = 1 + 8 = 9
\][/tex]
3. Calculate the square root of the discriminant:
[tex]\[
\sqrt{9} = 3
\][/tex]
4. Substitute these values back into the formula:
[tex]\[
x = \frac{1 \pm 3}{2 \cdot 1} = \frac{1 \pm 3}{2}
\][/tex]
So, our solutions are:
[tex]\[
x_1 = \frac{1 + 3}{2} = \frac{4}{2} = 2
\][/tex]
[tex]\[
x_2 = \frac{1 - 3}{2} = \frac{-2}{2} = -1
\][/tex]
Thus, the correct substitution and steps are shown in:
[tex]\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}
\][/tex]
The correct option is:
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)}
