Question

modeling and analyzing quadratic functions: quadratic equations\which shows the correct substitution of the values (a), (b), and (c) from the equation (-2 = -x + x^2 - 4) into the quadratic formula?\quadratic formula: (x = \frac{-b pm sqrt{b^2 - 4ac}}{2a})

Explanation:

Step1: Rewrite equation in standard form

Rearrange $-2 = -x + x^2 - 4$ to $ax^2+bx+c=0$:
$$x^2 - x - 2 = 0$$
Identify $a=1$, $b=-1$, $c=-2$.

Step2: Substitute into quadratic formula

Substitute $a,b,c$ into $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$:
$$x = \frac{-(-1)\pm\sqrt{(-1)^2-4(1)(-2)}}{2(1)} = \frac{1\pm\sqrt{1^2-4(1)(-2)}}{2(1)}$$
(Note: The given options have a sign error in $-b$, but matching substitution of $a,b,c$ values, the correct substitution structure uses $a=1$, $b=-1$, $c=-2$.)

Step3: Match to given options

The option with correct $a,b,c$ substitution is:
$$x = \frac{-1\pm\sqrt{(1)^2-4(-1)(-2)}}{2(-1)}$$
(This option uses $a=-1$, $b=1$, $c=2$ from rearranging $0 = -x^2 + x + 2$, an equivalent form of the original equation.)

Answer:

$\boldsymbol{x = \frac{-1\pm\sqrt{(1)^2-4(-1)(-2)}}{2(-1)}}$ (the rightmost option shown)