The Discriminant and Nature of Roots

The discriminant: a preview of the answer

Before solving a quadratic equation, the discriminant $\Delta = b^2 - 4ac$ tells you how many real solutions to expect — without actually solving.

The three cases

Case 1: $\Delta > 0$ — Two distinct real roots. The parabola crosses the x-axis at two different points. 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 just touches the x-axis at its vertex. Example: $x^2 - 4x + 4 = 0$ has $\Delta = 16 - 16 = 0$, root $x = 2$ (double root).

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

How to compute the discriminant

For $ax^2 + bx + c = 0$:

$\Delta = b^2 - 4ac$

Example: For $3x^2 + 2x - 5 = 0$: $\Delta = (2)^2 - 4(3)(-5) = 4 + 60 = 64 > 0$ → two real roots.

Why it works

The discriminant lives under the square root in the quadratic formula: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$. If $\Delta > 0$, $\sqrt{\Delta}$ is real and the $\pm$ gives two values. If $\Delta = 0$, the $\pm$ doesn't matter ($\pm 0 = 0$). If $\Delta < 0$, $\sqrt{\Delta}$ isn't real.

Exam tip

Many exam questions ask "how many real solutions?" or "for what values of $k$ does the equation have equal roots?" These are discriminant questions — set $\Delta = 0$ or $\Delta > 0$ and solve for $k$.

Try it in the visualization

Drag $a$, $b$, $c$ sliders. The discriminant value updates live, the classification appears, and the parabola visually shows the corresponding case. Try to make $\Delta$ exactly zero — watch the parabola become tangent to the x-axis.