Discriminant Calculator

Background.

Before you solve a quadratic, it helps to know what kind of answer is waiting. Will the parabola cross the x-axis twice, graze it once, or miss it altogether? The discriminant, b² − 4ac, settles that from the three coefficients alone, with no square roots and no factoring.

It is the part of the quadratic formula that sits under the radical: x = (−b ± √(b² − 4ac)) ÷ 2a. A positive value has a real square root, so the ± produces two different real roots. Zero makes the square root vanish and both branches land on the same number, −b ÷ 2a. A negative value has no real square root, and the two roots become complex numbers, conjugates of each other. For x² − 7x + 10 = 0 the discriminant is 49 − 40 = 9, positive, so two real roots; they turn out to be 2 and 5.

The number carries more than its sign. Its square root, divided by |a|, is the distance between the two real roots. With whole-number coefficients, a perfect-square discriminant such as 9 means the roots are rational and the quadratic factors neatly over the integers, while a positive non-square such as 8 means the roots involve a surd. Divided by −4a, it gives the height of the parabola's vertex above or below the axis.

That is why it turns up in so many problems that never ask for the roots themselves: whether a thrown ball reaches a given height, what value of a parameter gives equal roots, whether a damped spring oscillates or settles. Enter a, b and c; you get the discriminant and a code for the kind of roots it implies. To find the roots, take the same coefficients to the quadratic formula calculator.

What is discriminant calculator?

For a quadratic equation ax² + bx + c = 0 with a ≠ 0, the discriminant is D = b² − 4ac. Its sign sorts the roots into three cases. D > 0: two distinct real roots. D = 0: one real root of multiplicity two, often called a repeated or double root, where the quadratic is a perfect square a(x − r)². D < 0: no real roots, but two complex conjugate roots of the form p ± qi.

The name says what it does: it discriminates between those three behaviours. The same idea extends to every polynomial. A polynomial's discriminant is an expression in its coefficients that equals zero exactly when the polynomial has a repeated root, and the quadratic's is the only one short enough to memorise. For the cubic ax³ + bx² + cx + d it is already b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd.

The discriminant ignores the scale of the equation in one useful way. Multiply every coefficient by the same non-zero number k and D is multiplied by k², which can never change its sign, so 2x² − 14x + 20 = 0 has the same kind of roots as x² − 7x + 10 = 0.

How to use this calculator.

1. 01Rearrange the equation so one side is zero and the terms run ax² + bx + c. For 3x² = 5x − 2, that means 3x² − 5x + 2 = 0, so a = 3, b = −5 and c = 2.
2. 02Enter a, b and c, including their signs; a missing term has a coefficient of 0, and a bare x² has a = 1. Coefficient a cannot be zero, because then the equation is linear and has no discriminant.
3. 03Read the discriminant, then the root-nature code beside it: 0 for two distinct real roots, 1 for one repeated real root, 2 for a pair of complex conjugate roots.
4. 04If the coefficients are whole numbers and D is a perfect square, expect rational roots and try factoring; otherwise go straight to the quadratic formula, which needs √D.

The formula.

The formula comes from completing the square. Divide ax² + bx + c = 0 by a, move c/a across, and add (b/2a)² to both sides: (x + b/2a)² = (b² − 4ac) ÷ 4a². The left side is a square, so it can be zero or positive but never negative for real x. The right side has denominator 4a², which is always positive, so the sign of the whole right side is the sign of b² − 4ac. If it is positive, x + b/2a can be either of two square roots and there are two real solutions; if it is zero, only x = −b/2a works; if it is negative, no real x does.

The same rearrangement shows what the size of D means. The two roots are −b/2a ± √D/2a, so they sit symmetrically either side of x = −b/2a, at a distance √D/2|a| each way, and their difference is √D/|a|. Writing the quadratic in vertex form, a(x + b/2a)² − D/4a, shows that the vertex height is k = −D/4a. For an upward-opening parabola (a > 0), a positive D puts the vertex below the axis, so the curve must cross it twice.

A third route explains why D detects repeated roots in particular. A repeated root of f(x) = ax² + bx + c is also a root of its derivative, f′(x) = 2ax + b, which is zero only at x = −b/2a. Substituting that value gives f(−b/2a) = −D/4a, which is zero exactly when D is. The resultant of f and f′, the general algebraic test for a shared root, works out to −a(b² − 4ac).

When b is even, a shortcut saves arithmetic: writing b = 2h, D/4 = h² − ac. For x² − 6x + 5, h = −3 and h² − ac = 9 − 5 = 4, so D = 16.

Two points about the output. Decimal coefficients are handled as typed, so 0.1x² + 0.3x + 0.225 gives a discriminant of exactly 0 rather than a tiny binary-arithmetic leftover. And any discriminant whose size is below 10⁻⁹ is classified as a repeated root: x² + 1 × 10⁻¹⁰ has D = −4 × 10⁻¹⁰, strictly negative, yet is coded 1. For coefficients that small, rescale the equation first.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Stewart, J. (2015). Calculus: Early Transcendentals, 8th ed. Cengage Learning. ISBN 978-1285741550.
2. [2]NIST Digital Library of Mathematical Functions (2024). "Elementary Algebra." §1.11. https://dlmf.nist.gov/1.11
3. [3]Strang, G. (2016). Introduction to Linear Algebra, 5th ed. Wellesley-Cambridge Press. ISBN 978-0980232776.
4. [4]Boyce, W.E. and DiPrima, R.C. (2017). Elementary Differential Equations and Boundary Value Problems, 11th ed. Wiley. ISBN 978-1119382874.
5. [5]Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra, 3rd ed. Wiley. ISBN 978-0471433347.