# How do you find the discriminant and how many solutions does y= x^2 + 3x have?

May 8, 2015

The discriminant of an expression $a {x}^{2} + b x + c$ is
$\Delta = {b}^{2} - 4 a c$
(extracted from the quadratic formula for solutions
$x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

Extend the quadratic in the given equation so it looks like the general form:
$y = \left(1\right) {x}^{2} + 3 x + 0$

$\Delta = {3}^{2} - 4 \left(1\right) \left(0\right) = 9$

If you look back to the quadratic formula you can see:

$\Delta \left\{\begin{matrix}< 0 \rightarrow \text{no Real solutions" \\ =0 rarr "exactly 1 solution" \\ >0 rarr "2 solutions}\end{matrix}\right.$

In our case $\Delta > 0$
so $y = {x}^{2} + 3 x$ has 2 solutions