Quadratic Discriminant Calculator

Formula

Discriminant = b × b-4 × a × c

Worked example

With a: 1, b: -3, c: 2, discriminant is 1.

Calculation method

The rule is Discriminant = b × b-4 × a × c.

For ax² + bx + c = 0, the discriminant b² − 4ac determines the nature of the real roots: positive means two distinct real roots, zero means one repeated real root, and negative means no real roots.

For this quadratic discriminant example, changing a from 1 to 2 while keeping the other inputs fixed changes discriminant from 1 to -7.

Reading discriminant

The discriminant b² − 4ac controls the number of real roots. The plus and minus branches can return two different roots. In the worked values on this page, a 1, b -3, c 2 produce discriminant 1.

Practical use

Use the calculation to solve or check a quadratic equation from known coefficients. The specific task here is to find discriminant from a, b, c.

Limits and common mistakes

The coefficient a must be nonzero. A negative discriminant has no real roots, though complex roots exist outside this real-number tool. Verify a before interpreting discriminant; a value measured on another basis would answer a different question.

Why the relationship works

For ax² + bx + c = 0, the discriminant b² − 4ac determines whether real roots can be calculated. A positive value gives two distinct real roots, zero gives a repeated root, and a negative value gives no real root. The plus and minus formula branches must use the same coefficients and the same square-root term. The quadratic discriminant rule uses a, b, c in Discriminant = b × b-4 × a × c; read each symbol as the field named on this page.

Interpreting another set of values

For quadratic discriminant, the fields labeled a, b, c are the known quantities. The calculator solves for discriminant using Discriminant = b × b-4 × a × c. Interpret the answer in the context of the entered equation or coordinate system. A root, signed slope, intercept, distance, and remainder have different meanings and different valid domains. When comparing two calculations, keep their units and input definitions identical.

For a second quadratic discriminant check, keep the other default inputs fixed and change b from -3 to -1. Discriminant changes from 1 to -7. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

A quadratic needs a nonzero leading coefficient. The displayed root branches describe real results only when the discriminant is nonnegative. A rounded root may not make the polynomial exactly zero after substitution; use a reasonable tolerance when checking decimal output. The reported discriminant is 1 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.