Which answer shows the conjugate pair solutions to the equation x2−3x+8=−2 ?(1 point)

Answer 1

To find the conjugate pair solutions to the equation $x^2-3x+8=-2$, we first rearrange the equation:

$$x^2-3x+8+2=0$$

This simplifies to:

$$x^2-3x+10=0$$

Now, we can use the quadratic formula to find the solutions:

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

Here, $a=1$, $b=-3$, and $c=10$. Plugging in these values:

$$x=\frac{-(-3)\pm \sqrt{(-3{)}^2-4\cdot 1\cdot 10}}{2\cdot 1}$$

$$=\frac{3\pm \sqrt{9-40}}{2}$$ $$=\frac{3\pm \sqrt{-31}}{2}$$

Since the discriminant ($-31$) is negative, the solutions will be complex (conjugate pair). We express $\sqrt{-31}$ as $i\sqrt{31}$:

$$x=\frac{3\pm i\sqrt{31}}{2}$$

Thus, the conjugate pair solutions are:

$$x=\frac{3}{2}+\frac{i\sqrt{31}}{2}\text{and}x=\frac{3}{2}-\frac{i\sqrt{31}}{2}$$