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

Answer 1

To solve the equation \( x^2 - 5x + 10 = 0 \), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our equation, \( a = 1 \), \( b = -5 \), and \( c = 10 \). Plugging these values into the formula:

1. Calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = (-5)^2 - 4(1)(10) = 25 - 40 = -15 \]

2. Since the discriminant is negative, we will have complex solutions.

3. Plug these values into the quadratic formula:

\[ x = \frac{-(-5) \pm \sqrt{-15}}{2(1)} = \frac{5 \pm \sqrt{-15}}{2} \]

4. Simplify the square root of \(-15\):

\[ \sqrt{-15} = i\sqrt{15} \]

5. Now substitute back into the solution:

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

6. Split into real and imaginary parts:

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

Thus, the conjugate pair solutions of the equation \( x^2 - 5x + 10 = 0 \) are:

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