To solve the equation \( x^2 - 5x + 10 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are:
- \( a = 1 \)
- \( b = -5 \)
- \( c = 10 \)
Now, we first 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, the solutions will be complex. Now, we apply the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{-15}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{5 \pm \sqrt{-15}}{2} \]
To express the square root of a negative number in terms of complex numbers, we write:
\[ \sqrt{-15} = i\sqrt{15} \]
Substituting this back into the equation gives:
\[ x = \frac{5 \pm i\sqrt{15}}{2} \]
Breaking this down, we can write:
\[ x = \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]
Thus, the solutions for the equation \( x^2 - 5x + 10 = 0 \) are
\[ \frac{5}{2} \pm \frac{i\sqrt{15}}{2} \]