The Discriminant: How Many Solutions?

The discriminant: a preview of the answer

Before solving a quadratic $ax^2 + bx + c = 0$, the discriminant $\Delta = b^2 - 4ac$ tells you how many real solutions to expect.

The three cases

Case 1: $\Delta > 0$ — Two distinct real roots. The parabola crosses the x-axis twice. Example: $x^2 - 5x + 6 = 0$ has $\Delta = 25 - 24 = 1 > 0$, roots $x = 2$ and $x = 3$.

Case 2: $\Delta = 0$ — One repeated root. The parabola touches the x-axis at its vertex. Example: $x^2 - 4x + 4 = 0$ has $\Delta = 0$, root $x = 2$ (double root).

Case 3: $\Delta < 0$ — No real roots. The parabola doesn't reach the x-axis. Example: $x^2 + 1 = 0$ has $\Delta = -4$, no real solutions.

How to compute it

For $ax^2 + bx + c = 0$: $\Delta = b^2 - 4ac$. Example: $3x^2 + 2x - 5 = 0$: $\Delta = 4 + 60 = 64 > 0$ → two real roots.

Why it works

$\Delta$ lives under the square root in the quadratic formula: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$. Positive $\Delta$ → real square root → two solutions. Zero → one. Negative → no real square root.

Exam tip

Questions like "for what values of $k$ does $x^2 + kx + 9 = 0$ have equal roots?" → set $\Delta = 0$: $k^2 - 36 = 0$, so $k = \pm 6$.

Try it in the visualization

Drag $a$, $b$, $c$. The discriminant value, classification, and parabola update live. Try making $\Delta$ exactly zero — watch the parabola become tangent to the x-axis.