Why does a positive discriminant have two different cases?

Because whether it is a perfect square decides whether the roots are rational. √9 is 3, so a discriminant of 9 gives roots that are fractions and a quadratic that factorises with rational coefficients. √8 is irrational, so a discriminant of 8 gives a conjugate pair of surds that no rational factorisation reaches. Both are two real roots; only one of them can be found by factorising. Among quadratics with integer coefficients from −10 to 10 that have real roots, only 24.6% are the factorisable kind.

What happens if a is zero?

There is no quadratic, so there is no discriminant. bx + c = 0 is a straight line with one root at −c ÷ b, and the question "how many real roots" has a different answer for a different reason. The trap is that b² − 4ac still evaluates when a is zero — it gives b², a perfectly ordinary number that describes nothing. A calculator that does not check will show it without comment, which is why this page refuses instead.

What does a negative discriminant actually mean?

That the parabola never meets the x-axis. There are still two solutions, but they are complex conjugates — the same real part with opposite imaginary parts, written p ± qi. Nothing has failed: an equation with no real solution is a perfectly good state of affairs, and it is often the answer you want. A ball thrown that never reaches a given height gives a negative discriminant, and the correct conclusion is that it does not reach it.

How do you tell whether the discriminant is a perfect square?

Take its square root and see whether the result is rational. With integer coefficients the discriminant is an integer, so the test is whether it is a square number — 0, 1, 4, 9, 16 and so on. With fractional coefficients the discriminant is a fraction, and it is a perfect square when the numerator and denominator are both squares after cancelling. Doing this on a decimal is unreliable: √8 shows as 2.8284271, and no number of decimal places proves it is irrational.

What is the connection between the discriminant and the graph?

The sign of the discriminant is the number of times the parabola crosses the x-axis. Two crossings for positive, one touch at the vertex for zero, none for negative. Equivalently, it says where the vertex sits relative to the axis: for an upward parabola a vertex below the axis gives two crossings, on it gives one, above it gives none. The discriminant and the vertex are two ways of saying the same thing.

Does scaling the equation change the discriminant?

Yes, but not the roots. Doubling a, b and c multiplies the discriminant by four, because b² picks up a factor of four and 4ac does too. The roots are unchanged, since multiplying an equation by a non-zero constant cannot move its solutions. So the discriminant is not an intrinsic property of the roots — its sign is, and its being a perfect square is, but its size depends on how the equation was written.

Why compute the discriminant exactly rather than as a decimal?

Because near zero the sign is fragile, and the sign is the whole classification. When b² and 4ac are close, subtracting them in floating point loses accuracy exactly where it matters most. Over a family of quadratics whose b² and 4ac differ by only a few units, computing the discriminant in doubles got its sign wrong a quarter of the time — turning no real roots into one repeated root. The failure is not a wrong digit but a wrong answer to the question being asked.

Can the discriminant help before I decide how to solve?

That is its most useful role. Work it out first and you know immediately whether factorising is even possible — a perfect square means yes, anything else means the quadratic formula or completing the square. A negative value means stopping before you waste effort looking for real roots that are not there. Checking a single expression before choosing a method is faster than discovering halfway through that the method cannot work.

What about the discriminant of a cubic?

Cubics and higher polynomials have discriminants too, and they play the same role — telling you about the nature of the roots without finding them. The cubic discriminant is a considerably longer expression in the four coefficients, and its sign distinguishes three distinct real roots from one real root with two complex ones. The idea generalises; the formula does not stay convenient. This page is for quadratics, where b² − 4ac is short enough to be worth memorising.

Discriminant Calculator

The discriminant of a quadratic ax² + bx + c is b² − 4ac, the part of the quadratic formula that sits under the square root. Its sign decides how many real roots there are before any root is worked out: positive gives two, zero gives one, negative gives none. Everything else in the formula is unaffe...

A positive discriminant is two cases, not one. If it is a perfect square the roots are rational and the quadratic factorises; if it is not, the roots are a conjugate pair of surds that no rational factorisation will ever produce. Among quadratics with integer coefficients from −10 to 10 that have real roots at all, only 24.6% have a perfect-square discriminant — so most solvable quadratics cannot be factorised, and a decimal answer hides which kind you are looking at. If a is zero there is no quadratic and no discriminant, however willingly b² − 4ac evaluates.

THE COEFFICIENTS

For ax² + bx + c = 0. Whole numbers, decimals and fractions all work.

x² − 3x + 2 = 0

D = 1 · TWO RATIONAL ROOTS

x = 2, x = 1

2 and 1

Two real roots, and the discriminant is a perfect square, so both are rational and the quadratic factorises with rational coefficients.

FACTORISED

1(x − 2)(x − 1)

x = 2 → 0 ✓

x = 1 → 0 ✓

DISCRIMINANT

positive — two real roots

A PERFECT SQUARE

yes

so the roots are rational

FACTORISES

yes

over the rationals

ROOTS PUT BACK

2 of 2

each gives exactly zero

WHAT IS HAPPENING

1Work out b² − 4ac

(-3)² − 4(1)(2) = 9 − 8 = 1

This single number decides how many real roots there are, before any root is calculated. It is the part of the quadratic formula that sits under the square root sign, and everything else in the formula is unaffected by its sign.

2Positive, and a perfect square

√1 = 1 → x = 2 and x = 1

A positive discriminant that is the square of a rational number is the case where the roots come out rational — and that is precisely when the quadratic can be factorised with rational coefficients. Any other positive value gives roots that no such factorisation reaches.

3Where the parabola sits

vertex at (3/2, -1/4), opening upward

With a positive the parabola opens upward, so the vertex is its lowest point. A vertex below the axis means two crossings, on the axis means one, and above it means none — which is the discriminant restated in geometry.

Why the sign is the whole story. The quadratic formula reads (−b ± √D) ÷ 2a, and everything outside the square root is the same whatever D turns out to be. A positive D adds and subtracts a real amount, giving two crossings. A zero D adds and subtracts nothing, so the two roots land on the same point. A negative D asks for the square root of a negative number, and the parabola never reaches the axis at all. Nothing else in the formula is affected.

Exact fractions throughout · rational roots are substituted back into the equation as written

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter a, b and c for ax² + bx + c = 0. Whole numbers, decimals and fractions such as 3/4 all work, and the equation is written back to you so you can check it is the one you meant.
2
Read the discriminant and its sign. Positive means two real roots, zero means one repeated root, negative means none — and that is settled before any root is calculated.
3
Check whether it is a perfect square. That is the difference between rational roots you can factorise and irrational ones you cannot, and it is the distinction a decimal answer conceals.
4
Look at the parabola. Two crossings, one touch, or a curve that misses the axis entirely — the same three cases as the sign, drawn rather than asserted.

WORKED EXAMPLE

Take x² − 3x + 2 = 0. The discriminant is (−3)² − 4(1)(2) = 9 − 8 = 1. Positive, so there are two real roots. It is also a perfect square, so those roots are rational: (3 ± 1) ÷ 2 gives 2 and 1. The quadratic factorises as (x − 2)(x − 1), and substituting either root back gives exactly zero. The parabola crosses the axis twice. Now 2x² − 4x + 1 = 0. The discriminant is 16 − 8 = 8. Positive again, so two real roots again — but 8 is not a perfect square. The roots are (4 ± √8) ÷ 4, which simplifies to 1 ± √2 ÷ 2, roughly 1.707 and 0.293. They are perfectly exact and perfectly real, and no factorisation with rational coefficients produces them. Both quadratics have positive discriminants and only one of them can be factorised, which is the distinction worth carrying. Now x² + 2x + 1 = 0. The discriminant is 4 − 4 = 0. The ± in the formula is adding and subtracting nothing, so the two roots collapse onto one: x = −1. The quadratic is (x + 1)², and the parabola touches the axis at its vertex without crossing. Now x² + 2x + 5 = 0. The discriminant is 4 − 20 = −16. Negative, so no real roots at all. The formula asks for the square root of a negative number, and the two solutions are the complex conjugates −1 ± 2i. The parabola sits entirely above the axis. Nothing has gone wrong: the equation has no real solution, and that is the answer. Finally the case that catches calculators. Take x² + 200000000x + 10000000000000001. Here b² is 4 × 10¹⁶ and 4ac is 4 × 10¹⁶ + 4, so the discriminant is exactly −4 and there are no real roots. Computed in double precision the coefficient c cannot even be held exactly, and the discriminant comes out as 0 — one repeated root, which is the wrong answer to the wrong question. The sign is the entire classification, and near zero it is the first thing floating point loses.

REFERENCE RULES

Rule	What it says	Why
The discriminant	D = b² − 4ac	The part of the quadratic formula under the square root. Its sign decides everything.
Positive	two distinct real roots	The parabola crosses the x-axis twice.
Zero	one repeated root	The parabola touches the axis at its vertex and does not cross.
Negative	no real roots	The parabola misses the axis entirely. Two complex conjugate solutions.
Positive and a square	rational roots	Only then does the quadratic factorise with rational coefficients.
Positive, not a square	a conjugate pair of surds	Real and exact, but no rational factorisation reaches them.
If a = 0	there is no discriminant	It is not a quadratic. b² − 4ac still evaluates, and means nothing.
The roots	(−b ± √D) ÷ 2a	The ± is the whole reason two roots exist, and D decides what it adds.
The vertex	x = −b ÷ 2a	Halfway between the roots when they exist, whatever the discriminant.
Sum of the roots	−b ÷ a	True even when the roots are complex. The imaginary parts cancel.
Product of the roots	c ÷ a	Also true for complex roots, since conjugates multiply to a real number.
A perfect square trinomial	D = 0	x² + 2x + 1 and its relatives. The factorisation is a single squared bracket.
Scaling the equation	changes D, not the roots	Doubling a, b and c multiplies D by four and leaves every root where it was.
Near zero	the sign is fragile	When b² and 4ac are close, a floating-point subtraction can lose the sign entirely.

FREQUENTLY ASKED QUESTIONS

How many real roots a quadratic has, before you calculate any of them. It is b² − 4ac, the expression under the square root in the quadratic formula. Positive means the ± adds and subtracts a real amount, so there are two distinct roots. Zero means it adds and subtracts nothing, so the two roots coincide. Negative means the square root has no real value, so there are no real roots at all.

·b² − 4ac, the part under the square root
·Positive: two distinct real roots
·Zero: one repeated root
·Negative: no real roots

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Last updated: August 6, 2026 · Exact fractions throughout, never floating point · A positive discriminant is two cases, depending on whether it is a perfect square.