Question

Solve the equation x^2 - 5x + 10 = 0. Write your conjugate pair solution(s) as two values separated by a \pm sign.

Answer 1

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

\[
\frac{5}{2} \pm \frac{i\sqrt{15}}{2}
\]