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For a quadratic equation, $ax^2 + bx +c$, the discriminant is: $Δ = b^2 - 4ac$.

For a quadratic equation, $ax^2 + bx +c$, the discriminant is: $Δ = b^2 - 4ac$.

The discriminant is the part of the quadratic solution, $x = {-b ± √{b^2 - 4ac}}/{2a}$, under the root sign. If the discriminant is negative, the root has no solution in ℝ.

The discriminant determines how many solutions for $x$ there are when the function equals zero (i.e. how many x-axis intercepts there are).

$Δ > 0$ : 2 solutions for x when the function equals zero. The parabola crosses the x-axis two times.

$Δ = 0$ : 1 solution for x. The parabola touches the x-axis without crossing it.

$Δ > 0$ : 0 solutions for x. The parabola does not touch or cross the x-axis.