# How many solutions are there to the equation 0 = 3x^2 - 10x - 5?

Oct 20, 2015

compare this equation with a${x}^{2}$ +bx+c=0
a= 3, b=-10, c=-5
Discriminant= ${B}^{2}$ -4ac
= ${\left(- 10\right)}^{2}$ - 4*3(-5)
= 100+60
=160 which is more than 0
So it has two solutions.

Oct 20, 2015

Two

#### Explanation:

For a quadratic in the general form:
$\textcolor{w h i t e}{\text{XXX}} a {x}^{2} + b x + c = 0$
the discriminant
$\textcolor{w h i t e}{\text{XXX}} \Delta = {b}^{2} - 4 a c$
indicates the number of solutions:
$\Delta \left\{\begin{matrix}< 0 & \text{no solutions" \\ = 0 & "exactly one solution" \\ > 0 & "two solutions}\end{matrix}\right.$

In this case $a = 3$, $b = - 10$, and $c = - 5$
so $\Delta = {\left(- 10\right)}^{2} - 4 \left(3\right) \left(- 5\right) = 160 > 0$
$\Rightarrow$ two solutions.

Further, the discriminant is part of the quadratic formula that gives the actual solutions:
$\textcolor{w h i t e}{\text{XXX}} x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

In this case
$\textcolor{w h i t e}{\text{XXX}} x = \frac{5 \pm 2 \sqrt{10}}{2}$