The Discriminant

The discriminant of a quadratic equation determines how many and what type of roots the equation has.

Definition

For the quadratic $ax^2 + bx + c = 0$ , the discriminant is:

\Delta = b^2 - 4ac

The discriminant determines the nature and number of roots.

$\Delta > 0$ : Two distinct real roots. The parabola crosses the x-axis at two points.
$\Delta = 0$ : Exactly one real root (a repeated root). The parabola is tangent to the x-axis.
$\Delta < 0$ : No real roots. The parabola does not intersect the x-axis.

Tip

Without computing the full quadratic formula, the discriminant alone tells you how many solutions to expect.

Classify roots of 3x² − 2x + 5 = 0

$\Delta = (-2)^2 - 4(3)(5) = 4 - 60 = -56$

Since $\Delta < 0$ , there are no real roots.

Real world: will a ball clear the wall?

A ball follows the path $h = -t^2 + 6t + 1$ . We ask: does it ever reach $h = 10$ ?

Set $-t^2 + 6t + 1 = 10$ → $t^2 - 6t + 9 = 0$ .

$\Delta = 36 - 36 = 0$ . The discriminant is exactly zero — the ball just grazes height 10 m at one precise moment ( $t = 3$ s), then falls back.

Real world: can a business break even?

A company's profit model is $P = -2x^2 + 8x - 10$ (thousands of euros, $x$ = units sold in hundreds).

For break-even, set $P = 0$ : $\Delta = 64 - 80 = -16 < 0$ .

Because $\Delta < 0$ , there are no real solutions — the business never breaks even with this pricing model.

Real world: resonant frequencies in a circuit

An engineer models a circuit with $L\omega^2 - R\omega + \frac{1}{C} = 0$ . The discriminant $\Delta = R^2 - \frac{4L}{C}$ tells them:

$\Delta > 0$ : two distinct resonant frequencies exist.
$\Delta = 0$ : critically damped — one resonant frequency.
$\Delta < 0$ : overdamped — no real resonant frequency; the signal simply decays.

For x² − 4x + 4 = 0, compute Δ and classify the roots

Show answer

Δ = 16 − 16 = 0. One repeated real root: x = 2

For x² − 4x + 3 = 0, compute Δ and classify the roots

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Δ = 16 − 12 = 4 > 0. Two distinct real roots.