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, \(a = 1\), \(b = -5\), and \(c = 10\).
First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-5)^2 - 4(1)(10) = 25 - 40 = -15 \]
Since the discriminant is negative, the equation has complex solutions.
Now we plug the values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-15}}{2(1)} = \frac{5 \pm \sqrt{-15}}{2} \]
We can express \(\sqrt{-15}\) as \(i\sqrt{15}\), where \(i\) is the imaginary unit. Now, we have:
\[ x = \frac{5 \pm i\sqrt{15}}{2} \]
Thus, the two solutions can be expressed as:
\[ x = \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]
So the conjugate pair solution(s) can be written as:
\[ \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]