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Discriminant Calculator

Step 1 / 4 Discriminant of ax² + bx + c = b² − 4ac
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Before you run the full quadratic formula, a single expression tells you everything about the roots you are going to get. The discriminant b^2 - 4ac is positive when there are two real roots, zero when there is exactly one (a repeated root), and negative when the roots are a complex conjugate pair. This tool computes it from your a, b, c coefficients and explains what the value means.

How to compute the discriminant

  1. 1

    Enter the coefficients

    Type a, b and c from the standard form ax^2 + bx + c = 0. Decimals and negatives are accepted.

  2. 2

    Read the discriminant

    The tool shows D = b^2 - 4ac and the numeric result.

  3. 3

    Interpret the value

    See whether the equation has two distinct real roots, one repeated root, or two complex roots.

What the sign means

Discriminant Root type Graph behavior
D > 0 Two distinct real roots Parabola crosses the x-axis twice
D = 0 One real root (repeated/double) Parabola touches the x-axis at one point
D < 0 Two complex conjugate roots Parabola never crosses the x-axis

Worked examples

x^2 - 5x + 6 = 0: D = 25 - 24 = 1. Positive, so two real roots. Using the formula: x = (5 +/- 1) / 2, giving x = 3 and x = 2.

x^2 - 4x + 4 = 0: D = 16 - 16 = 0. Zero, so one repeated root. x = 4/2 = 2.

x^2 + 2x + 5 = 0: D = 4 - 20 = -16. Negative, so complex roots. x = (-2 +/- sqrt(-16))/2 = -1 +/- 2i.

Beyond the sign: the exact value

The discriminant also tells you whether the roots are rational. For integer a, b, c:

  • D is a perfect square (0, 1, 4, 9, 16, 25, …): roots are rational.
  • D is positive but not a perfect square: roots are irrational, involving a square root.
  • D is negative: roots are non-real complex numbers.

Knowing this ahead of time tells you whether the equation factors cleanly over the integers.

Related discriminants

The idea generalizes to higher-degree polynomials. A cubic has a discriminant involving b, c, d and constant terms that classifies the three roots into all-real-distinct, real-with-repeat, or one-real-two-complex. The calculator here handles only the quadratic case; for cubics and quartics use a general algebra solver.

Tips

  • Always check signs. A stray minus sign on c flips the discriminant dramatically.
  • Simplify fractions first. If your coefficients are rationals, clear denominators before computing D to keep the arithmetic manageable.
  • Double-check real-world problems. In projectile motion, a negative discriminant means the projectile never reaches the target, not that you made an arithmetic mistake.

Frequently Asked Questions

Not directly. The b^2 - 4ac formula is specific to degree-2 polynomials. Higher-degree discriminants exist but use different formulas. If your equation is not quadratic, reduce it first (substitution, factoring) until you reach a quadratic piece.

In a projectile problem, zero discriminant is the boundary case where the projectile just barely reaches the target with one tangent path. In a circuit problem, it is the critically damped case. Physically it is the threshold between two different behaviors.

The calculation runs on the server with double-precision arithmetic, and the result is shown rounded to 4 decimal places. Inputs beyond 15 significant digits may be approximated.

No. The coefficients are sent to our server so the result can update as you type, and in the step-by-step mode they are also carried in the page link. They are not stored or logged.

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