The discriminant calculator above computes b2 − 4ac for any quadratic and then does the part that actually matters: it interprets the number. A discriminant is not an answer in itself. It is a classifier that tells you, before you solve anything, how many real roots exist, whether they are rational or irrational, and whether the parabola touches the x-axis, crosses it twice, or misses it entirely.
This page is deliberately narrow. If you want the roots themselves, Arb Digital's quadratic equation solver gives you x1 and x2 directly. This tool is for the question that comes before that one: what kind of solution should you expect, and what does the discriminant reveal that the roots alone do not?
What This Discriminant Calculator Does
Enter a, b and c. The calculator returns the discriminant, the number of distinct real roots, and a classification of the root type — rational, irrational, repeated, or complex conjugate. It also gives the square root of the discriminant, which is the quantity that appears under the radical in the quadratic formula, and the axis of symmetry at x = −b/(2a), which is where the parabola turns regardless of whether it meets the axis.
The working line shows the substitution with your numbers, so b2 − 4ac is written out term by term rather than presented as a finished figure.
How to Use It
- Put the quadratic in standard form, ax2 + bx + c = 0, with everything on one side and zero on the other.
- Read off a, b and c with their signs. In 2x2 + 5x − 3 = 0 they are 2, 5 and −3.
- Enter the three values. A missing linear term means b = 0; a missing constant means c = 0.
- Press Calculate to see the discriminant and its interpretation together.
- Read the root type to decide whether the equation will factor neatly or produce a surd.
The Formula and the Three Cases
The discriminant is the expression under the square root in the quadratic formula, x = (−b ± √(b2 − 4ac)) / (2a). Because everything else in that formula is an ordinary real calculation, the entire character of the solution is decided by the sign of one quantity:
- Δ > 0 — two distinct real roots. The square root is a positive real number, so the ± produces two different answers. The parabola crosses the x-axis twice.
- Δ = 0 — one repeated real root. The square root is zero, so both branches collapse to x = −b/(2a). The parabola touches the axis at its vertex without crossing.
- Δ < 0 — a complex conjugate pair. The square root of a negative number is imaginary, giving roots of the form p ± qi. The parabola never meets the x-axis.
The formal definition, along with its generalisation to higher-degree polynomials, is given in the Wolfram MathWorld entry on the polynomial discriminant.
The Fourth Case Most Pages Skip: Rational vs. Irrational Roots
Textbooks usually stop at three cases. There is a fourth distinction hiding inside the first one, and it is the reason the discriminant is genuinely useful rather than merely descriptive.
When a, b and c are integers and Δ is a perfect square, the roots are rational — which means the quadratic factors over the integers and you can solve it by inspection instead of reaching for the formula. When Δ is positive but not a perfect square, the roots are irrational surds that arrive as a conjugate pair, p ± √q, and no amount of factoring by grouping will find them.
This is a practical time-saver. Faced with 6x2 + 11x − 35, computing Δ = 121 + 840 = 961 = 312 tells you in one step that a clean factorisation exists and is worth hunting for. Faced with a discriminant of 962, you know immediately that no integer factor pair will work and you should go straight to the formula. The calculator above performs that perfect-square test whenever your coefficients are integers.
A Fully Worked Example
Take 2x2 + 5x − 3 = 0, so a = 2, b = 5 and c = −3.
Substitute: Δ = b2 − 4ac = 52 − 4(2)(−3) = 25 − (−24) = 25 + 24 = 49. Note the sign trap in the middle: because c is negative, −4ac becomes a positive contribution, and losing that sign is the single most common arithmetic error in this calculation.
Now interpret. Δ = 49 is positive, so there are two distinct real roots. It is also a perfect square, 72, so those roots are rational and the quadratic factors over the integers. Indeed 2x2 + 5x − 3 = (2x − 1)(x + 3), giving x = 1/2 and x = −3. The axis of symmetry is x = −b/(2a) = −5/4 = −1.25, which sits exactly halfway between 0.5 and −3, as it must.
How the Discriminant Behaves When You Move the Parabola
Changing c shifts the parabola vertically, and that is the cleanest way to see the discriminant do its job. Start with x2 − 4x + 3, whose discriminant is 16 − 12 = 4, positive, with roots at 1 and 3. Raise c to 4 and the discriminant becomes 16 − 16 = 0: the curve has risen just far enough that its lowest point rests on the axis, giving the repeated root x = 2. Raise c to 5 and the discriminant is −4: the curve has lifted clear of the axis and there are no real roots at all.
The value of c that makes Δ = 0 is exactly b2/(4a), and that is a genuinely useful piece of algebra. It is how you find the value of an unknown parameter that makes a quadratic a perfect square, and it is the standard route into questions that ask for the condition on k such that an equation has equal roots. Set the discriminant to zero and solve for k. The relationship between the discriminant, the vertex and the completed-square form is laid out in the Wolfram MathWorld entry on the quadratic equation, which is worth reading alongside this page if you want the algebra rather than the summary.
Discriminants Beyond the Quadratic
Every polynomial has a discriminant, and it always answers the same question: does this polynomial have a repeated root? A discriminant of zero means yes, for a quadratic, a cubic or anything higher. What changes is how much extra information the sign carries.
For a cubic the discriminant is −4p3 − 27q2 in depressed form, and a positive value means three distinct real roots while a negative value means one real root and a complex pair. For quartics and beyond, the sign no longer pins down the case uniquely — a negative quartic discriminant tells you there are exactly two real roots, but a positive one leaves two possibilities open, either four real roots or none. The quadratic is the only case where one number gives a complete answer, which is part of why it is taught first and used so heavily.
Where the Discriminant Shows Up in Applications
Projectile problems use it constantly. Asking whether a thrown object ever reaches a given height is asking whether a quadratic in time has a real root, and the discriminant answers that without solving for the time. In geometry, substituting a line into the equation of a circle produces a quadratic whose discriminant tells you whether the line cuts the circle in two points, touches it as a tangent, or misses it — the tangency condition is precisely Δ = 0. Engineering uses it in the same spirit to test whether a damped oscillator is underdamped, critically damped or overdamped, because those three regimes correspond exactly to the three signs of the discriminant of the characteristic equation.
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Browse Free Tools Talk To UsCommon Mistakes to Avoid
- Losing the sign on −4ac. When c is negative, −4ac adds to b2 rather than subtracting from it.
- Forgetting to square the whole of b. If b = −6 then b2 = 36, not −36.
- Reading the equation before rearranging it. In x2 = 5x − 4 the value of b is −5 and c is 4, not 5 and −4.
- Saying a negative discriminant means "no solution". It means no real solution. Two complex roots still exist and still satisfy the equation.
- Assuming any positive discriminant factors nicely. Only a perfect-square discriminant guarantees rational roots; otherwise the answer is a surd.
Related Free Tools From Arb Digital
Once the discriminant tells you what to expect, get the actual roots from the quadratic equation solver. The square root calculator handles √Δ when it is irrational, the exponent calculator helps with the squaring step, and the slope calculator is useful for the line-meets-curve tangency problems described above. See the full free online tools hub for the rest.
Frequently Asked Questions
It is the expression b squared minus four a c, taken from under the square root in the quadratic formula. Its sign determines how many real roots the equation has and what type they are.
There are no real roots. The two solutions are complex conjugates of the form p plus q i and p minus q i, and the parabola never touches the x-axis.
The equation has exactly one repeated real root, located at minus b over two a. The parabola touches the x-axis at its vertex without crossing it, and the quadratic is a perfect square.
If a, b and c are integers and the discriminant is a perfect square, the roots are rational and the quadratic factors over the integers. If it is positive but not a perfect square, the roots are irrational surds.
No. It only classifies them. To get the actual values you still need the quadratic formula or factoring, which is what our quadratic equation solver does.
Yes. Setting the discriminant equal to zero and solving for the unknown gives the condition for equal roots, which is a standard exam question and a common tangency condition in geometry.
Yes, and so does every polynomial. A discriminant of zero always signals a repeated root, but only for the quadratic does the sign alone fully determine the number of real roots.
This calculator is provided for study and reference. Results are computed in floating point, so confirm any classification by substituting your values back into b squared minus four a c by hand.
