9.4 Quadratic equations and complex numbers

In the unit on quadratic equations and complex numbers, we saw that a quadratic equation can have two answers, one answer, or no answers .

We can now modify this third case. In cases where we described “no answers” there are actually two answers, but both are complex! This is easy to see if you remember that we found “no answers” when the discriminant was negative —that is, when the quadratic formula gave us a negative answer in the square root.

As an example, consider the equation:

${\mathrm{2x}}^2+\mathrm{3x}+5=0$

The quadratic equation gives us:

$x=\frac{-3\pm \sqrt{3^2-4(2)(5)}}{4}$ $=$ $\frac{-3\pm \sqrt{-\text{31}}}{4}$

This is the point where, in the “old days,” we would have given up and declared “no answer.” Now we can find two answers—both complex.

$=\frac{-3}{4}\pm \frac{\sqrt{\text{31}}\sqrt{-1}}{4}$ $=$ $\frac{-3}{4}\pm \frac{\sqrt{\text{31}}}{4}i$

So we have two answers. Note that the two answers are complex conjugates of each other—this relationship comes directly from the quadratic formula.