Sure, let's go through the steps to figure out the correct substitution for the values [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the equation [tex]\(-2 = -x + x^2 - 4\)[/tex] into the quadratic formula.
First, we need to rewrite the equation in standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
Given equation:
[tex]\[ -2 = -x + x^2 - 4 \][/tex]
Rearrange it to:
[tex]\[ 0 = x^2 - x - 4 - 2 \][/tex]
Simplify:
[tex]\[ 0 = x^2 - x - 6 \][/tex]
So now we see that the quadratic equation in standard form is:
[tex]\[ x^2 - x - 6 = 0 \][/tex]
Here,
[tex]\[ a = 1, \quad b = -1, \quad c = -6 \][/tex]
Now we substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
From the results, we know the correct substitution looks like:
[tex]\[x=\frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}\][/tex]
To match this against our possible choices:
[tex]\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-6)}}{2(1)}\][/tex]
Putting [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -6\)[/tex] into the formula:
1. [tex]\(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)}\)[/tex] is almost correct but has the incorrect value for [tex]\(c\)[/tex] substitution (should be [tex]\(-6\)[/tex] not [tex]\(-4\)[/tex]).
2. [tex]\(x = \frac{-1 \pm \sqrt{1^2 - 4(-1)(-4)}}{2(-1)}\)[/tex] tries to change negative substitution incorrectly, and signs are mismatched.
3. [tex]\(x = \frac{-1 \pm \sqrt{1^2 - 4(-1)(-2)}}{2(-1)}\)[/tex] rearranges quadratic factors wrongly.
4. [tex]\(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}\)[/tex] uses previous solutions to frame rightly.
The correct answer and substitution from all this would be:
[tex]\[x=\frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}\][/tex]
