To find the conjugate pair solutions to the equation \( x^2 - 3x + 8 = -2 \), we first need to rewrite it in standard form:
\[ x^2 - 3x + 10 = 0 \]
Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -3 \), and \( c = 10 \).
Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 10 = 9 - 40 = -31 \]
Since the discriminant is negative, we will have complex solutions. Now, we can substitute into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{-31}}{2 \cdot 1} = \frac{3 \pm i\sqrt{31}}{2} \]
Thus, the solutions are:
\[ x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]
Now, let's compare this with the given options:
- First option: \( \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \) (matches)
- Second option: \( \frac{3}{2} \pm \frac{i\sqrt{15}}{2} \) (does not match)
- Third option: \( -\frac{3}{2} \pm \frac{i\sqrt{31}}{2} \) (does not match)
- Fourth option: \( \frac{3}{2} \pm \frac{i\sqrt{23}}{2} \) (does not match)
The first option shows the correct conjugate pair solutions to the equation:
\[ \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]