# How to use the discriminant to find out what type of solutions the equation has for x^2 - 8x + 3 = 0?

May 23, 2015

For a quadratic of the form
$a {x}^{2} + b x + c = 0$
the discriminant is
$\Delta = {b}^{2} - 4 a c$
where
$\Delta \left\{\begin{matrix}< 0 \rightarrow \text{no Real solutions" \\ =0 rarr "1 Real solution" \\ >0 rarr "2 Real solutions}\end{matrix}\right.$

Given
${x}^{2} - 8 x + 3 = 0$
$\Delta = {\left(- 8\right)}^{2} - 4 \left(1\right) \left(3\right) = 52$

Since $\Delta > 0$
this equation has 2 Real solutions.

(By the way, since $\sqrt{\Delta}$ is irrational, both solutions are also irrational.)