How this instrument works
The discriminant is the exact expression that sits under the square root in the quadratic formula, x = (−b ± √(b² − 4ac)) / 2a. Complete the square on ax² + bx + c = 0 and you land on (x + b/2a)² = (b² − 4ac) / 4a² — the discriminant is what's left once every other term has been squared away, and its sign decides whether that final square root is even a real number.
A positive Δ means the parabola crosses the x-axis at two separate points — two real roots, as with x² − 5x + 6 factoring into (x−2)(x−3). A Δ of exactly zero means the parabola is tangent to the axis, touching at one repeated root, as in x² − 4x + 4 = (x−2)². A negative Δ means the parabola never reaches the axis at all; the roots still exist, just as a complex-conjugate pair. The mathematician James Joseph Sylvester named this quantity 'the discriminant' in 1851, precisely because it discriminates between these three cases without any further arithmetic.
One property surprises people: multiply an entire equation by any nonzero constant k, so a, b, and c all scale by k, and the discriminant scales by k² — always positive — so its sign, and therefore the number of real roots, never changes. The one input that has to stay away from zero is a; set it to 0 and the x² term disappears, the equation stops being quadratic, and b² − 4ac no longer means anything. The most common slip by hand is losing the minus sign on 4ac when b itself is negative — squaring b = −5 correctly gives +25, but it's easy to subtract 4ac before squaring b and land on the wrong number entirely.
- Enter the coefficient of x² into the a (coefficient of x²) field — for a plain x² term, leave it at 1.
- Enter the coefficient of x into the b (coefficient of x) field, keeping the sign that appears in your equation (a −5x term means b = −5).
- Enter the constant term into the c (constant) field.
- Read Discriminant, b² − 4ac: its sign alone tells you how many real roots the quadratic has, before you ever apply the quadratic formula.
Worked example — x² − 5x + 6
Take the quadratic x² − 5x + 6 = 0, so a = 1, b = −5, and c = 6. The discriminant is Δ = b² − 4ac = (−5)² − 4(1)(6) = 25 − 24 = 1. Because 1 is strictly positive, the equation is guaranteed to have two distinct real roots before you ever apply the quadratic formula.
Those roots check out directly: x² − 5x + 6 factors as (x − 2)(x − 3), giving roots x = 2 and x = 3 — two separate, real, rational numbers, exactly as a positive discriminant promises. Plugging the same a, b, and c into the full formula gives x = (5 ± √1) / 2, which is (5 ± 1) / 2 = 3 or 2, confirming the discriminant's prediction without any extra work.
Questions
What does a negative discriminant mean?
It means the quadratic has no real roots — the parabola never touches the x-axis. The two solutions still exist, but only as a complex-conjugate pair, such as ±i for x² + 1 = 0, where Δ = 0² − 4(1)(1) = −4. Nothing is broken; the equation simply has no real-number solution.
How is the discriminant related to the quadratic formula?
It's the exact quantity under the square root in x = (−b ± √(b² − 4ac)) / 2a. Checking its sign first tells you whether that square root will be a positive real number, zero, or the square root of a negative — without computing the rest of the formula at all.
What happens when the discriminant equals zero?
The quadratic has exactly one repeated real root — the parabola's vertex sits right on the x-axis. For x² − 4x + 4 = 0, Δ = (−4)² − 4(1)(4) = 16 − 16 = 0, and the equation factors as a perfect square, (x − 2)², with its single root at x = 2.
Does the sign of a, b, or c change the discriminant's meaning?
No — only the combination b² − 4ac matters, and squaring b erases its sign entirely. Multiplying the whole equation by −1 flips a, b, and c but leaves b² − 4ac unchanged, since (−b)² = b² and (−a)(−c) = ac. The discriminant only cares about magnitude and the relationship between the three coefficients, not which ones happen to be negative.
Why is it called a 'discriminant'?
Because it discriminates — literally distinguishes — between the three possible root behaviors of a quadratic without solving for x first. A positive value signals two real roots, zero signals one repeated root, and a negative value signals a complex-conjugate pair. The name, and the general concept extended to any polynomial, come from 19th-century algebra.
Can the discriminant idea be used for cubic or higher equations?
Yes — every polynomial has a discriminant, though the formula grows quickly more complex past degree two. For a quadratic it is simply b² − 4ac; a cubic's discriminant already involves nine terms. This calculator handles only the quadratic case, ax² + bx + c.