Math July 13, 2026 · 8 Min Read

Discriminant Calculator – Guide & Formulas

Calculate the discriminant b² - 4ac of any quadratic equation. Determine if roots are real, repeated, or complex with step-by-step solutions.

Calculate the discriminant of any quadratic equation instantly with our free calculator. Enter a, b, and c to find b² - 4ac and classify the roots as real, repeated, or complex.

Key Takeaway

Use the free Discriminant Calculator to calculate the discriminant b² - 4ac of any quadratic equation. determine if roots are real, repeated, or complex with step-by-step solutions. Get instant results with step-by-step explanations.

How to Use the Discriminant Calculator

  1. Enter coefficient a (the x² term) — must be non-zero.
  2. Enter coefficient b (the x term).
  3. Enter coefficient c (the constant term).
  4. Review the discriminant value, root type, and actual roots when real.

The Formula

For the quadratic equation ax² + bx + c = 0, the discriminant is Δ = b² - 4ac. If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: two complex conjugate roots.

Variable Definitions

  • Δ: The discriminant value: b² - 4ac
  • a: Coefficient of the quadratic term (x²), must be non-zero
  • b: Coefficient of the linear term (x)
  • c: Constant term
  • x₁, x₂: The roots of the quadratic: x = (-b ± √Δ) / (2a)

Discriminant of x² - 5x + 6 = 0

Classify the roots and find their values.

  1. Step 1: Identify a = 1, b = -5, c = 6.
  2. Step 2: Compute Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1.
  3. Step 3: Δ = 1 > 0, so there are two distinct real roots.
  4. Step 4: Roots: x = (5 ± √1) / 2 = (5 ± 1) / 2. So x = 3 or x = 2.

Frequently Asked Questions

What is the discriminant of a quadratic equation?

The discriminant is the expression Δ = b² - 4ac, where a, b, and c are the coefficients of ax² + bx + c = 0. It determines the number and type of roots without fully solving the equation.

What does a positive discriminant mean?

When Δ > 0, the quadratic equation has two distinct real roots. The parabola crosses the x-axis at two different points.

What does a zero discriminant mean?

When Δ = 0, the equation has exactly one repeated (double) real root. The parabola touches the x-axis at exactly one point — its vertex.

What does a negative discriminant mean?

When Δ < 0, there are no real roots. Instead, the equation has two complex conjugate roots. The parabola never crosses the x-axis.

How does the discriminant relate to the quadratic formula?

The discriminant appears under the square root in the quadratic formula: x = (-b ± √Δ) / (2a). Its sign determines whether √Δ is real, zero, or imaginary.

Can the discriminant be used for higher-degree polynomials?

The discriminant concept extends to cubic and quartic equations, but the formula becomes more complex. For quadratics, Δ = b² - 4ac is the standard expression.

What if a = 0?

If a = 0, the equation is not quadratic — it becomes linear (bx + c = 0). The discriminant formula does not apply. Solve directly: x = -c/b.

Is a perfect square discriminant special?

Yes. When Δ is a perfect square (and a, b, c are integers), the quadratic factors neatly over the integers, giving rational roots.