The discriminant of a quadratic is the term under the square root in the quadratic formula:

When Δ<0, the square root has a negative content, and so the quadratic has no real solutions.

When Δ=0, the square root is zero, and the ±√Δ makes no difference, so there is only one real solution.

When Δ>0, √Δ is positive and so ±√Δ yields two real roots.

Example

For what value(s) of k does the quadratic 2x2−3x+k have 2 real roots?

The discriminant is Δ=(−3)2−4⋅2⋅k=9−8k.

There are two real solutions when Δ>0 so k<89​.