Discriminant Calculator
Find the discriminant of a quadratic, a cubic, or a two-variable conic equation — and see exactly what that one number tells you about the roots or the shape, with worked steps and a supporting visual.
Background
A discriminant is a single number, built only from a polynomial's (or curve's) coefficients, that reveals how many real roots exist and whether any of them repeat — all without solving anything. For a quadratic it decides how many times a parabola crosses the x-axis; for a cubic it decides whether all three roots are real or only one is; and for a general conic equation the same idea, applied to two variables, decides whether the curve is an ellipse, a parabola, or a hyperbola.
How to use this calculator
- Choose Quadratic for ax²+bx+c=0 — the classic b²−4ac, which tells you whether a parabola crosses the x-axis twice, touches it once, or never touches it. The result also shows how far your c is from the exact tangency value.
- Choose Cubic for ax³+bx²+cx+d=0 — a more elaborate formula that tells you whether all three roots are real, whether one repeats, or whether only one root is real. The visual marks the curve's turning points so you can literally see why it crosses once or three times.
- Choose Conic section for the general two-variable quadratic curve Ax²+Bxy+Cy²+Dx+Ey+F=0 — B²−4AC classifies the curve itself as an ellipse, a parabola, or a hyperbola. For an axis-aligned ellipse or circle, it also reports the eccentricity.
- Click Calculate to see the discriminant, a plain-language explanation of what it means, a supporting chart, and full step-by-step working.
- Try the Golden Ratio, Cube Roots of Unity, and Casus Irreducibilis quick examples — three famous equations where the discriminant tells a genuinely surprising story.
How the discriminant works
A discriminant is a single number computed only from a polynomial's (or curve's) coefficients that reveals how many roots are real, and whether any repeat, without ever solving for the roots themselves.
For a quadratic, b²−4ac tells you — before you even use the quadratic formula — whether the parabola crosses the x-axis twice, touches it once, or never touches it at all.
Sign is what matters, not magnitude: positive means distinct real roots, zero means a repeated root, and negative means the roots exist but only as a complex conjugate pair.
The idea generalizes to higher degrees: cubics, quartics, and beyond all have a discriminant that is exactly zero precisely when the polynomial has a repeated root. For a cubic, this is a direct consequence of geometry: a repeated root means the curve is exactly tangent to the x-axis at a turning point, instead of crossing cleanly through it.
The same core idea classifies conic sections: instead of asking "how many times does this cross an axis," B²−4AC asks "what shape does this two-variable quadratic curve trace" — ellipse, parabola, or hyperbola. Eccentricity is a related, more familiar number that also flows from the same A, B, C: 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, and above 1 for a hyperbola.
A discriminant of exactly zero is always a tangency or boundary condition — the precise knife-edge between two qualitatively different behaviors.
A cubic with a positive discriminant has three real roots, yet the classical Cardano formula for finding them algebraically still requires passing through complex numbers as an intermediate step — a famous quirk called the casus irreducibilis ("the irreducible case"). This calculator sidesteps that detour with a numerically robust real-arithmetic approach instead.
Formula & Equations Used
Quadratic discriminant: D = b² − 4ac, with roots x = (−b ± √D) / (2a)
Quadratic vertex link: yvertex = −D / (4a) — the discriminant is literally 4a times the vertex's height above (or below) the x-axis.
Tangency value of c: ctangent = b² / 4a — the exact value c would need to be for D to equal zero, holding a and b fixed.
Cubic discriminant: Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²
Cubic classification: Δ>0 gives three distinct real roots; Δ=0 gives a repeated real root; Δ<0 gives one real root and a complex conjugate pair.
Cubic turning points: where the derivative 3ax²+2bx+c=0, i.e. x = (−b ± √(b²−3ac)) / 3a — three real roots occur exactly when the local max is on one side of the x-axis and the local min is on the other.
Casus irreducibilis: when Δ>0, Cardano's radical formula for a real root necessarily involves the square root of a negative number partway through, even though the final answer is real.
Conic discriminant: B² − 4AC for Ax²+Bxy+Cy²+Dx+Ey+F=0
Conic classification: negative gives an ellipse (a circle when A=C and B=0); zero gives a parabola; positive gives a hyperbola.
Eccentricity (axis-aligned ellipse/circle, B=0): after centering, e = √(1 − (semi-minor/semi-major)²); a parabola always has e=1 regardless of coefficients.
Example Problems & Step-by-Step Solutions
These cover cases the Quick Examples chips above don't already demonstrate.
Example 1 — Solving for a tangency condition
For what value of k does x²+kx+9=0 have exactly one repeated real root?
Step: A repeated root means D=0: k²−4(1)(9)=0, so k²=36.
Result: k = ±6 — at exactly those two values, the parabola is tangent to the x-axis.
Example 2 — A negative quadratic discriminant
Find the discriminant and roots of x²+2x+5=0.
Step: D = 2²−4(1)(5) = 4−20 = −16, so the roots are complex: x = (−2 ± √−16)/2.
Result: x = −1 ± 2i — the parabola never touches the x-axis. Here c would need to drop all the way to 1 (b²/4a) to reach tangency, so this is comfortably in "no real roots" territory.
Example 3 — Three real roots that aren't consecutive integers
Classify x³−7x+6=0 using its discriminant.
Step: With a=1,b=0,c=−7,d=6: Δ=18(1)(0)(−7)(6)−4(0)³(6)+0²(−7)²−4(1)(−7)³−27(1)²(6)² = 1372−972 = 400.
Result: Δ=400>0, so all three roots are real: x = −3, 1, 2. This one happens to have rational roots, so it doesn't strictly need the casus irreducibilis detour — but any irreducible cubic with Δ>0 would.
Example 4 — A double root that isn't a full triple root
Classify x³−x²−x+1=0, which factors as (x−1)²(x+1).
Step: The discriminant formula gives Δ=0, matching the fact that x=1 is a repeated root.
Result: Δ=0 — roots are 1, 1, −1: the local turning point on that side sits exactly on the x-axis, which is what forces the repeat.
Example 5 — A rotated parabola
Classify x²−4xy+4y²+x=0 using the conic discriminant.
Step: A=1, B=−4, C=4, so B²−4AC = 16−16 = 0.
Result: Discriminant = 0, so it's a parabola — here (x−2y)²+x=0, tilted off the usual axes by that cross term. Its eccentricity is exactly 1, as it always is for a parabola.
Example 6 — Recognizing a circle in disguise
Classify x²+y²−6x+8y=0.
Step: A=1, B=0, C=1, so B²−4AC = 0−4 = −4 — negative, so it's an ellipse.
Result: Discriminant = −4, and since A=C and B=0, it's the special ellipse case: a circle, centered at (3,−4), with eccentricity exactly 0.
Frequently Asked Questions
Does a negative discriminant mean there's no solution?
Not "no solution" — just no real solution. The roots still exist as a complex conjugate pair; they simply don't correspond to a point where the graph crosses the real x-axis.
Why is the cubic discriminant formula so much messier than b²−4ac?
Algebraic complexity grows fast with degree. Both formulas are really the same underlying object — a resultant of the polynomial and its derivative — but that resultant has far more terms once the polynomial has four coefficients instead of three.
Is the conic discriminant B²−4AC really the same idea as b²−4ac?
Yes. Both come from the same quadratic-form structure; the conic version is just applied to a curve in two variables instead of a one-variable equation, which is why it classifies a shape instead of counting roots.
Can a cubic have a zero discriminant but no repeated root?
No — for a cubic, a discriminant of exactly zero always means at least two roots coincide. That's true by definition: the discriminant of any polynomial vanishes exactly when it has a repeated root.
What exactly is the casus irreducibilis?
It's the case where a cubic has three real roots (Δ>0) but is irreducible — no rational root exists to factor out. Cardano's classical radical formula still technically works, but it requires taking the cube root of a complex number along the way, even though every root that comes out the other end is real.
Why do the turning points determine how many times a cubic crosses zero?
A cubic's graph rises to +∞ on one end and falls to −∞ on the other (or vice versa). If its local max and local min sit on opposite sides of the x-axis, the curve has to swing across zero three separate times to get from one turning point to the other. If they're on the same side (or there are no turning points at all), it only needs to cross once.
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