To find the conjugate pair solutions to the equation \( x^2 - 3x + 8 = -2 \), we first rearrange the equation to standard form:
\[ x^2 - 3x + 10 = 0 \]
Next, we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, \( a = 1 \), \( b = -3 \), and \( c = 10 \). We can substitute these values into the formula.
Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 10 = 9 - 40 = -31 \]
Since the discriminant is negative, this indicates that the solutions will be complex.
Now substituting back 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 provided answer options:
- Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction
This corresponds to \( \frac{3}{2} \pm \frac{i \sqrt{31}}{2} \).
So the correct answer is:
Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction