# When is the discriminant of a quadratic function imaginary?

Oct 28, 2015

The discriminant of a quadratic function can only be imaginary if at least some of the coefficients of the quadratic are imaginary.

#### Explanation:

For a quadratic in the general form
$\textcolor{w h i t e}{\text{XXX}} y = a {x}^{2} + b x + c$

The discriminant is
$\textcolor{w h i t e}{\text{XXX}} {b}^{2} - 4 a c$

If the discriminant is negative (which might be what you intended to ask)
the square root of the discriminant is imaginary
and therefore the quadratic formula
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
gives imaginary values as roots for $y = 0$
This happens when the parabola does not touch or cross the X-axis.