Question

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

Answer 1

You first write out the discriminant `D=b^2-4ac`

Explanation:

We know the values of `a` and `b`, so:
`D=(-1)^2-4*2*(-c)=1+8c`

For two (unequal) real roots `D>0`, or:
`1+8c>0->8c> -1->c> -1/8`

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

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