Solve the equation x2−5x+10=0

. Write your conjugate pair solution(s) as two values separated by a ±
sign.

1 answer

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:

  • \( a = 1 \)
  • \( b = -5 \)
  • \( c = 10 \)

Now, plug these values into the formula:

  1. Calculate the discriminant \( b^2 - 4ac \): \[ (-5)^2 - 4 \times 1 \times 10 = 25 - 40 = -15 \]

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

  3. Now use the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{-15}}{2 \times 1} = \frac{5 \pm \sqrt{-15}}{2} \]

  4. We can express \( \sqrt{-15} \) as \( i\sqrt{15} \), where \( i \) is the imaginary unit.

  5. Therefore, the solutions can be simplified to: \[ x = \frac{5 \pm i\sqrt{15}}{2} \]

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

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

This can be expressed as:

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