Question

Which answer shows the conjugate pair solutions to the equation x^2 - 3x + 8 = -2?
A. (3)/(2) \pm (i √(15))/(2)
B. (3)/(2) \pm (i √(23))/(2)
C. -(3)/(2) \pm (i √(31))/(2)
D. (3)/(2) \pm (i √(31))/(2)

Answer 1

To find the conjugate pair solutions to the equation \( x^2 - 3x + 8 = -2 \), we first simplify the equation:

\[
x^2 - 3x + 8 + 2 = 0
\]

This simplifies to:

\[
x^2 - 3x + 10 = 0
\]

We can use the quadratic formula to solve for \( x \), which is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -3 \), and \( c = 10 \):

1. Calculate \( b^2 - 4ac \):

\[
b^2 = (-3)^2 = 9
\]
\[
4ac = 4 \cdot 1 \cdot 10 = 40
\]
\[
b^2 - 4ac = 9 - 40 = -31
\]

2. Substitute into the quadratic formula:

\[
x = \frac{3 \pm \sqrt{-31}}{2}
\]

3. Simplifying gives:

\[
x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2}
\]

Thus, the conjugate pair solutions are:

\[
\frac{3}{2} \pm \frac{i\sqrt{31}}{2}
\]

Looking at the provided options, the correct answer is:

**D. \(\frac{3}{2} \pm \frac{i\sqrt{31}}{2}\)**