# How do you find the discriminant and how many solutions does h(x)=x^2-2x-35 have?

May 10, 2015

Solutions to a quadratic of the form $a {x}^{2} + b x + c = 0$ are given by
the quadratic formula $\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The sub-expression within the square root determines the number (and type) of solutions; this sub-expression is called the "discriminant" and is typically expressed as:
$\Delta = {b}^{2} - 4 a c$
with the conditions
$\Delta \left\{\begin{matrix}< 0 \text{ there are no Real solutions" \\ =0" there is 1 Real solution" \\ >0" there are 2 Real solutions}\end{matrix}\right.$

Given $h \left(x\right) = {x}^{2} - 2 x - 35$

$\Delta = {\left(- 2\right)}^{2} - 4 \left(1\right) \left(- 35\right)$
$= 4 + 140 = 144$
$> 0$
so there are two Real solutions