To find the conjugate pair solutions to the equation \( x^2 - 3x + 8 = -2 \), we first rearrange the equation:
\[ x^2 - 3x + 8 + 2 = 0 \]
This simplifies to:
\[ x^2 - 3x + 10 = 0 \]
Now, we can use the quadratic formula to find the solutions:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -3 \), and \( c = 10 \). Plugging in these values:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 10}}{2 \cdot 1} \]
\[ = \frac{3 \pm \sqrt{9 - 40}}{2} \] \[ = \frac{3 \pm \sqrt{-31}}{2} \]
Since the discriminant (\( -31 \)) is negative, the solutions will be complex (conjugate pair). We express \( \sqrt{-31} \) as \( i\sqrt{31} \):
\[ x = \frac{3 \pm i\sqrt{31}}{2} \]
Thus, the conjugate pair solutions are:
\[ x = \frac{3}{2} + \frac{i\sqrt{31}}{2} \quad \text{and} \quad x = \frac{3}{2} - \frac{i\sqrt{31}}{2} \]
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