# What is the discriminant of y= -3x^2 - 4x - 3 and what does that mean?

Aug 12, 2015

-20

#### Explanation:

In the general form of a quadratic expression $f \left(x\right) = a {x}^{2} + b x + c$, the discriminant is $\Delta = {b}^{2} - 4 a c$. Comparing the given expression with the form, we get $a = - 3$, $b = - 4$, and $c = - 3$. Thus the discriminant is $\Delta = {\left(- 4\right)}^{2} - 4 \left(- 3\right) \left(- 3\right) = 16 - 36 = - 20$.

The general solution of the equation $f \left(x\right) = 0$ for such a quadratic expression is given by $x = \frac{- b \pm \sqrt{\Delta}}{2 a}$.

If the discriminant is negative, taking square root would give you imaginary values. In essence, we understand that there are no real solutions of the equation $f \left(x\right) = 0$. This means that the graph of $y = f \left(x\right)$ never cuts the x-axis. Since $a = - 3 < 0$, the graph is always below the x-axis.

Do note that we do have complex solutions, namely $x = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{- \left(- 4\right) \pm \sqrt{- 20}}{2 \left(- 3\right)} = \frac{- 4 \pm 2 \sqrt{5} i}{6} = - \frac{2}{3} \pm \frac{\sqrt{5} i}{3}$.