To solve the given equation [tex]\(x^2 - 3x = -8\)[/tex], let's follow a detailed step-by-step approach.
1. Rewrite the equation in standard quadratic form:
[tex]\[
x^2 - 3x + 8 = 0
\][/tex]
This is a standard quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 8\)[/tex].
2. Identify the coefficients:
[tex]\[
a = 1, \quad b = -3, \quad c = 8
\][/tex]
3. Use the quadratic formula:
The quadratic formula for solving [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute the coefficients into the formula:
[tex]\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}
\][/tex]
5. Simplify the expression inside the square root:
[tex]\[
(-3)^2 = 9
\][/tex]
[tex]\[
4ac = 4 \cdot 1 \cdot 8 = 32
\][/tex]
[tex]\[
b^2 - 4ac = 9 - 32 = -23
\][/tex]
6. Insert these values back into the quadratic formula:
[tex]\[
x = \frac{3 \pm \sqrt{-23}}{2}
\][/tex]
7. Simplify the square root of a negative number:
The term [tex]\(\sqrt{-23}\)[/tex] can be rewritten using imaginary units as [tex]\(\sqrt{-23} = i\sqrt{23}\)[/tex].
[tex]\[
x = \frac{3 \pm i\sqrt{23}}{2}
\][/tex]
8. Express the solutions:
There are two complex solutions:
[tex]\[
x = \frac{3 + i\sqrt{23}}{2} \quad \text{and} \quad x = \frac{3 - i\sqrt{23}}{2}
\][/tex]
Now let's compare these solutions with the given answer choices:
- [tex]\(x = \frac{3 \pm i \sqrt{29}}{2}\)[/tex]
- [tex]\(x = \frac{3 \pm i \sqrt{23}}{2}\)[/tex]
- [tex]\(x = \frac{-3 \pm i \sqrt{29}}{2}\)[/tex]
- [tex]\(x = \frac{-3 \pm i \sqrt{23}}{2}\)[/tex]
The correct answer, based on our detailed solution, is:
[tex]\[
x = \frac{3 \pm i\sqrt{23}}{2}
\][/tex]
Thus, the matching answer from the provided choice is:
[tex]\[
\boxed{x = \frac{3 \pm i \sqrt{23}}{2}}
\][/tex]
