Last updated: 2026-05-19

Definition

The discriminant of a quadratic equation ax² + bx + c = 0 is the expression D = b² − 4ac. It is the single number that determines whether the equation has two distinct real roots, one repeated real root, or two complex conjugate roots, without first solving the equation. Its sign distinguishes the three cases, so it serves as the initial check before computing the roots themselves with the quadratic formula.

The formula: D = b² − 4ac

Computing the discriminant by hand takes three steps:

1. Square the middle coefficient: b²
2. Multiply the other two: 4 × a × c
3. Subtract: D = b² − 4ac

No square roots or fractions are involved. The sign of D alone determines the nature of the roots.

Geometric interpretation

The quadratic equation ax² + bx + c = 0 asks where the parabola y = ax² + bx + c crosses the x-axis. The discriminant answers that question geometrically:

When D = 0 the vertex of the parabola sits exactly on the x-axis, so x = −b / 2a is simultaneously the axis of symmetry and the only real root.

Worked examples

Example 1 — D > 0 (two distinct real roots)

Equation: x² − 5x + 6 = 0 (a = 1, b = −5, c = 6)

Step 1: b² = (−5)² = 25
Step 2: 4ac = 4 × 1 × 6 = 24
Step 3: D = 25 − 24 = 1

Since D = 1 > 0, there are two distinct real roots. Solving confirms x = 2 and x = 3, which are the x-intercepts of the parabola y = x² − 5x + 6.

Example 2 — D = 0 (one repeated real root)

Equation: x² − 6x + 9 = 0 (a = 1, b = −6, c = 9)

Step 1: b² = 36
Step 2: 4ac = 36
Step 3: D = 36 − 36 = 0

The single root is x = −(−6) / (2 × 1) = 3. This is a perfect square: x² − 6x + 9 = (x − 3)². The parabola's vertex touches the x-axis at x = 3.

Example 3 — D < 0 (two complex conjugate roots)

Equation: x² + x + 1 = 0 (a = 1, b = 1, c = 1)

Step 1: b² = 1
Step 2: 4ac = 4
Step 3: D = 1 − 4 = −3

Since D = −3 < 0, there are no real roots. The roots are complex: $x = (-1 \pm \sqrt{3}\,i) / 2 \approx -0.5 \pm 0.866i$. $\sqrt{|D|} = \sqrt{3} \approx 1.732$ is the imaginary magnitude shown by this calculator.

Relationship to the quadratic formula

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the square root is exactly the discriminant D. When D > 0, $\sqrt{D}$ is real and positive, giving two distinct values of x. When D = 0, $\sqrt{D} = 0$ and the ± produces only one value. When D < 0, $\sqrt{D} = \sqrt{|D|} \cdot i$ is imaginary, so the formula yields complex numbers. The discriminant is the part of the formula that controls which of the three outcomes applies — computing it first saves time when only the type of roots is needed rather than their exact values.

Edge case: a = 0

If a = 0, the equation ax² + bx + c = 0 loses its x² term and becomes the linear equation bx + c = 0. The formula D = b² − 4ac still produces a number (b²), but it has no meaning for a linear equation. This calculator requires a ≠ 0 and displays an error message when a = 0. For a linear equation, the solution is simply x = −c / b (assuming b ≠ 0).