Discriminant Calculator

About the Discriminant Calculator

Before solving a polynomial equation, it helps to know what kind of solutions to expect. The discriminant is a single number, computed from the coefficients, that answers that question. For a quadratic it tells you whether there are two real roots, one repeated root or no real roots at all, and whether real roots are rational. For a cubic it tells you whether there are three real roots or just one.

This discriminant calculator works for both quadratics and cubics. Enter the coefficients and it computes the discriminant exactly, using fractions if your coefficients are decimals, explains what the sign means for the roots and, for a quadratic, lists the roots themselves as a check.

How to Use the Discriminant Calculator

Choose quadratic for ax² + bx + c = 0 or cubic for ax³ + bx² + cx + d = 0.

Enter the coefficients a, b, c and, for a cubic, d. The leading coefficient a cannot be zero.

Coefficients can be negative and can include decimals.

The Formulas

  quadratic:  Δ = b² − 4ac
              roots x = (−b ± √Δ) / 2a
  cubic:      Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²

Step-by-Step Example: A Quadratic

2x² + 3x − 5 = 0.

  Δ = 3² − 4 × 2 × (−5) = 9 + 40 = 49
  Δ > 0 and 49 = 7², so there are two distinct rational roots
  x = (−3 ± 7) / 4  →  x = 1 and x = −2.5

Step-by-Step Example: A Cubic

x³ − 6x² + 11x − 6 = 0, whose roots are 1, 2 and 3.

  18abcd  = 18 × 1 × (−6) × 11 × (−6) = 7128
  −4b³d   = −4 × (−216) × (−6)       = −5184
  b²c²    = 36 × 121                 = 4356
  −4ac³   = −4 × 1 × 1331            = −5324
  −27a²d² = −27 × 1 × 36             = −972
  Δ = 7128 − 5184 + 4356 − 5324 − 972 = 4 > 0  →  three distinct real roots

Reading a Quadratic Discriminant

Δ > 0: two distinct real roots. The parabola crosses the x-axis twice. If the coefficients are rational and Δ is a perfect square, the roots are rational and the quadratic factors over the integers or rationals; otherwise the roots are irrational, as for x² + x − 1, where Δ = 5 and the roots are (−1 ± √5)/2.

Δ = 0: one repeated real root, x = −b/2a. The parabola just touches the x-axis at its vertex. x² − 4x + 4 = (x − 2)² is an example.

Δ < 0: no real roots. The roots are a pair of complex conjugates, −b/2a ± (√−Δ/2a)i, and the parabola stays entirely above or below the x-axis. x² + 2x + 5 has Δ = −16 and roots −1 ± 2i.

Reading a Cubic Discriminant

Every cubic with real coefficients has at least one real root, because its graph runs from minus infinity to plus infinity. The discriminant says what happens to the other two. Δ > 0 means three distinct real roots. Δ = 0 means a repeated root, and all roots are real; x³ − 3x² + 3x − 1 = (x − 1)³ is an extreme case with a triple root. Δ < 0 means one real root and a pair of complex conjugate roots, as for x³ + 1, where Δ = −27.

Note that the sign convention for cubics is the opposite of what students sometimes expect from the quadratic case: a negative cubic discriminant does not mean no real roots, only that two of the three are complex.

Why the Discriminant Is Useful

The discriminant lets you classify roots without solving. That is handy when sketching graphs, checking whether a quadratic can be factored neatly, or deciding how many times a line meets a curve: substituting the line into the curve's equation gives a quadratic, and its discriminant tells you whether the line misses, touches or crosses. In physics and engineering, the sign of a discriminant decides whether a system is overdamped, critically damped or underdamped.

Where the Formula Comes From

Completing the square on ax² + bx + c = 0 gives (x + b/2a)² = (b² − 4ac)/4a². The left side is a square, so it can never be negative for real x. If the right side is positive there are two real values of x + b/2a, if it is zero there is one, and if it is negative there are none. The numerator of that right side, b² − 4ac, is the discriminant, which is why it decides everything.

For a polynomial in general, the discriminant equals a power of the leading coefficient times the product of the squared differences of all pairs of roots. If two roots are equal, one of the differences is zero, so the discriminant is zero. That is why a zero discriminant always signals a repeated root, for polynomials of any degree.

Discriminants With Decimal Coefficients

When the coefficients are decimals, the calculator converts them to exact fractions before computing, so 0.5x² + 0.1x − 0.3 gives Δ = 61/100 exactly rather than a rounded value. That matters at the boundary: a discriminant that should be exactly zero might otherwise come out as a tiny positive or negative number and suggest the wrong kind of roots.

Discriminant and the Graph

The discriminant also describes the graph. For y = ax² + bx + c, a positive Δ means the parabola crosses the x-axis at two points, a zero Δ means its vertex sits exactly on the axis, and a negative Δ means the whole curve lies on one side of the axis: above it when a is positive, below it when a is negative. For a cubic, a positive Δ means the curve has a local maximum and minimum on opposite sides of the axis, so it crosses three times.

Understanding Your Result

The headline gives the discriminant, exact, as a whole number or fraction.

The roots line explains what the sign and value mean.

The equation line restates the equation.

The roots found line lists the quadratic's roots, real or complex, to six significant figures.

When Should You Use This Calculator?

Use it to classify the roots of a quadratic or cubic before solving.

Use it to check whether a quadratic factors with whole numbers.

Use it to find how many times a line and a parabola intersect.

Common Mistakes

Dropping the sign of c. With c = −5, −4ac becomes +40.

Squaring a negative b incorrectly. (−4)² = 16, not −16.

Reading a negative cubic discriminant as no real roots. A cubic always has at least one.

Using a = 0. Then the equation is not a quadratic or cubic.