To solve the equation \( x^2 - 3x + 8 = -2 \), we first move \(-2\) to the left side, resulting in:
\[ x^2 - 3x + 10 = 0 \]
Next, we can use the quadratic formula to find the solutions, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -3 \), and \( c = 10 \). We can substitute these values into the formula:
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Calculate \( b^2 - 4ac \): \[ (-3)^2 - 4(1)(10) = 9 - 40 = -31 \]
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Now substitute the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{-31}}{2 \cdot 1} = \frac{3 \pm \sqrt{-31}}{2} \]
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Since \( \sqrt{-31} = i\sqrt{31} \), we can rewrite the solutions as: \[ x = \frac{3 \pm i\sqrt{31}}{2} \]
This means the solutions can be expressed as: \[ x = \frac{3}{2} \pm \frac{i\sqrt{31}}{2} \]
So, the correct answer showing the conjugate pair solutions is:
Start Fraction 3 over 2 End Fraction plus or minus Start Fraction i Start Root 31 End Root over 2 End Fraction.
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