# What is the discriminant of 2x^2-7x-4=0 and what does that mean?

Jul 22, 2015

The discriminant of $2 {x}^{2} - 7 x - 4 = 0$ is $81$ and this means that there are 2 Real solutions for $x$ to this equation.

#### Explanation:

The discriminant for a quadratic equation in the form
$\textcolor{w h i t e}{\text{XXXX}}$$a {x}^{2} + b x + c = 0$
is
$\textcolor{w h i t e}{\text{XXXX}}$$\Delta = {b}^{2} - 4 a c$

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

For the given equation: $2 {x}^{2} - 7 x - 4 = 0$

$\Delta = {\left(- 7\right)}^{2} - 4 \left(2\right) \left(- 4\right)$
$\textcolor{w h i t e}{\text{XXXX}}$$= 49 + 32$
$\textcolor{w h i t e}{\text{XXXX}}$$= 81$
which tells us that there are 2 Real solutions

Jul 22, 2015

Solve $y = 2 {x}^{2} - 7 x - 4 = 0$

#### Explanation:

$D = {d}^{2} = {b}^{2} - 4 a c = 49 + 32 = 81$ --> $d = \pm 9$

This mean there are 2 real roots (2 x-intercepts). They are given by the formula:
$x = - \frac{b}{2 a} \pm \frac{d}{2 a}$
$x = \frac{7}{4} \pm \frac{9}{4}$
$x 1 = \frac{16}{4} = 4$
$x 2 = - \frac{2}{4} = - \frac{1}{2}$