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

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Answer 1 · 2017-04-03T12:13:13

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

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]}

How do use the discriminant to find all values of c for which the equation 2x^2-x-c=02x^2-x-c=0 has two real roots?