To solve the quadratic equation \(x^2 - 5x + 10 = 0\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -5\), and \(c = 10\).
First, we need to calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 1 \cdot 10 = 25 - 40 = -15 \]
Since the discriminant is negative, we will have complex solutions. Now we can substitute into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-15}}{2 \cdot 1} = \frac{5 \pm \sqrt{-15}}{2} \]
We can write \(\sqrt{-15} = i\sqrt{15}\) (where \(i\) is the imaginary unit).
Now, substituting this back into the equation gives us:
\[ x = \frac{5 \pm i\sqrt{15}}{2} \]
Thus, the solutions are:
\[ x = \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]
So the final answer, written as a conjugate pair, is:
\[ \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]