To solve the equation \( x^2 - 3x + 8 = -2 \), we first rearrange it into standard quadratic form:
\[ x^2 - 3x + 8 + 2 = 0 \] This simplifies to:
\[ x^2 - 3x + 10 = 0 \]
Next, we can use the quadratic formula to find the solutions to this equation. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 - 3x + 10 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -3 \)
- \( c = 10 \)
Now we can substitute these values into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 10}}{2 \cdot 1} \] This simplifies to:
\[ x = \frac{3 \pm \sqrt{9 - 40}}{2} \] \[ x = \frac{3 \pm \sqrt{-31}}{2} \] Since the discriminant \( -31 \) is negative, the solutions will be complex. We can express the square root of a negative number as a complex number:
\[ \sqrt{-31} = i\sqrt{31} \]
Substituting this back into our formula gives:
\[ x = \frac{3 \pm i\sqrt{31}}{2} \]
Thus, the conjugate pair solutions to the equation 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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