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Solve [tex]\( x^2 - 3x = -8 \)[/tex]

A. [tex]\( x = \frac{3 \pm i \sqrt{29}}{2} \)[/tex]

B. [tex]\( x = \frac{3 \pm i \sqrt{23}}{2} \)[/tex]

C. [tex]\( x = \frac{-3 \pm i \sqrt{29}}{2} \)[/tex]

D. [tex]\( x = \frac{-3 \pm i \sqrt{23}}{2} \)[/tex]


Sagot :

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]
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