To choose an equation that would be used to solve [tex]\(0 = -x^2 + 10x - 8\)[/tex], let us follow these steps:
1. Identify the type of equation:
The given equation [tex]\(0 = -x^2 + 10x - 8\)[/tex] is a quadratic equation. Quadratic equations often have solutions that can be expressed using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
2. Rewrite the equation in standard form:
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. For the given equation:
[tex]\[
-x^2 + 10x - 8 = 0
\][/tex]
Here, the coefficients are:
[tex]\[
a = -1, \quad b = 10, \quad c = -8
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is calculated as:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
For the given coefficients:
[tex]\[
\Delta = 10^2 - 4 \cdot (-1) \cdot (-8) = 100 - 32 = 68
\][/tex]
4. Calculate the roots:
Using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting in the values:
[tex]\[
x = \frac{-10 \pm \sqrt{68}}{-2}
\][/tex]
Simplifying further, we get two roots (solutions):
[tex]\[
x_1 = \frac{-10 + \sqrt{68}}{-2}, \quad x_2 = \frac{-10 - \sqrt{68}}{-2}
\][/tex]
Simplifying these, we get:
[tex]\[
x_1 = 5 - \sqrt{17}, \quad x_2 = 5 + \sqrt{17}
\][/tex]
5. Formulating the factorized form:
The roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] can be used to generate the equation in the form [tex]\((x - x_1)(x - x_2) = 0\)[/tex]. Rewriting it:
[tex]\[
(x - (5 - \sqrt{17}))(x - (5 + \sqrt{17})) = 0
\][/tex]
6. Recognize the quadratic in vertex form:
With known roots, we can express the equation in vertex form, which looks like [tex]\((x - h)^2 = k\)[/tex].
The roots are symmetrically distributed around their average, which is the vertex:
[tex]\[
h = \frac{x_1 + x_2}{2} = \frac{(5 - \sqrt{17}) + (5 + \sqrt{17})}{2} = 5
\][/tex]
The constant term [tex]\(k\)[/tex] is derived from calculating:
[tex]\[
k = (x_1 - h)^2 = (x_2 - h)^2 = (\sqrt{17})^2 = 17
\][/tex]
Therefore, the correct equation that could be used to solve [tex]\(0 = -x^2 + 10x - 8\)[/tex] is:
[tex]\[
(x - 5)^2 = 17
\][/tex]
