Sure, let's solve the quadratic equation [tex]\((x - 8)^2 - 13(x - 8) + 30 = 0\)[/tex] using the substitution method step-by-step.
1. Substitution:
Start by substituting [tex]\( u = x - 8 \)[/tex]. This substitution simplifies our equation into a more standard quadratic form.
[tex]\[
(x - 8)^2 - 13(x - 8) + 30 = u^2 - 13u + 30 = 0
\][/tex]
2. Quadratic Equation:
Now we need to solve the quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[
u^2 - 13u + 30 = 0
\][/tex]
3. Quadratic Formula:
The general form of the quadratic equation is [tex]\( au^2 + bu + c = 0 \)[/tex]. Here, [tex]\(a = 1\)[/tex], [tex]\(b = -13\)[/tex], and [tex]\(c = 30\)[/tex]. Using the quadratic formula:
[tex]\[
u = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Substituting the values:
[tex]\[
u = \frac{{-(-13) \pm \sqrt{{(-13)^2 - 4 \cdot 1 \cdot 30}}}}{2 \cdot 1}
\][/tex]
Simplifying the discriminant:
[tex]\[
u = \frac{{13 \pm \sqrt{{169 - 120}}}}{2}
\][/tex]
[tex]\[
u = \frac{{13 \pm \sqrt{49}}}{2}
\][/tex]
[tex]\[
u = \frac{{13 \pm 7}}{2}
\][/tex]
4. Solving for [tex]\(u\)[/tex]:
This gives us two values for [tex]\(u\)[/tex]:
[tex]\[
u = \frac{{13 + 7}}{2} = \frac{20}{2} = 10
\][/tex]
[tex]\[
u = \frac{{13 - 7}}{2} = \frac{6}{2} = 3
\][/tex]
5. Back Substitution:
We revert back to the original variable [tex]\(x\)[/tex], using the substitution [tex]\( u = x - 8 \)[/tex]:
[tex]\[
u = 10 \Rightarrow x - 8 = 10 \Rightarrow x = 18
\][/tex]
[tex]\[
u = 3 \Rightarrow x - 8 = 3 \Rightarrow x = 11
\][/tex]
6. Solutions:
Therefore, the solutions to the equation [tex]\((x - 8)^2 - 13(x - 8) + 30 = 0\)[/tex] are:
[tex]\[
x = 11 \quad \text{and} \quad x = 18
\][/tex]
So, the correct option is:
[tex]\[
\boxed{x = 11 \text{ and } x = 18}
\][/tex]
