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

$$\frac{5}{2}\pm \frac{i\sqrt{15}}{2}$$