To solve the quadratic equation \( x^2 - 5x + 10 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, the coefficients are:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 10 \)
Now, we can substitute these values into the quadratic formula.
First, calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 10 = 25 - 40 = -15 \]
Since the discriminant is negative, the solutions will be complex. Now we can calculate the values of \( x \):
\[ x = \frac{-(-5) \pm \sqrt{-15}}{2 \cdot 1} = \frac{5 \pm \sqrt{-15}}{2} \]
We can express \( \sqrt{-15} \) as \( i\sqrt{15} \), where \( i \) is the imaginary unit. Thus, we have:
\[ x = \frac{5 \pm i\sqrt{15}}{2} \]
So the conjugate pair solutions can be written as:
\[ x = \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]
Therefore, the final solution is:
\[ \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]
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