Which answer shows the conjugate pair solutions to the equation x^2 - 3x + 8 = -2?

Question

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. $\frac{3}{2}\pm \frac{i\sqrt{31}}{2}$