# How do use the discriminant to find all values of c for which the equation 2x^2-x-c=0 has two real roots?

Apr 3, 2017

You first write out the discriminant $D = {b}^{2} - 4 a c$

#### Explanation:

We know the values of $a$ and $b$, so:
$D = {\left(- 1\right)}^{2} - 4 \cdot 2 \cdot \left(- c\right) = 1 + 8 c$

For two (unequal) real roots $D > 0$, or:
$1 + 8 c > 0 \to 8 c > - 1 \to c > - \frac{1}{8}$

Below is the graph of the borderline case, where $D = 0$, and the parabole will be $2 {x}^{2} - x - \left(- \frac{1}{8}\right)$
graph{2x^2-x+1/8 [-10, 10, -5, 5]}

You will see, that as $c > - \frac{1}{8}$ the parabole will sink.
Here $c = + 1$ so the parabole will be $2 {x}^{2} - x - \left(+ 1\right)$
graph{2x^2-x-1 [-10, 10, -5, 5]}