The discriminant of any quadratic — root nature, both roots, vertex, and sum and product of roots.
How to use this tool
The coefficients of ax² + bx + c = 0. a must not be zero — otherwise it is not a quadratic.
The big number is b² − 4ac. Its sign alone tells you how many real roots exist.
Both roots (real or complex), the turning point, and the sum and product of the roots sit underneath.
Under Advanced options, enter an x value to get f(x) for the same quadratic.
Formula & worked example
Worked example: x² − 3x + 2 = 0 has D = 9 − 8 = 1, so two distinct real roots: (3 ± 1) ÷ 2 = 2 and 1. Check: sum = 3 = −(−3)/1 and product = 2 = 2/1, exactly as the formulas promise.
Limitations — read this before you rely on it
The discriminant classifies roots over the real numbers; complex roots come in conjugate pairs when coefficients are real. If coefficients are not real, or a = 0, the classification does not apply — this calculator refuses a = 0 rather than misclassifying it.
Frequently asked questions
Whether a quadratic has real roots, without solving it: D > 0 means two distinct real roots, D = 0 means one repeated root, and D < 0 means no real roots (two complex ones).
D = b² − 4ac from ax² + bx + c = 0. For x² − 3x + 2: D = (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
When D = 0, both roots coincide at −b/2a — the parabola just touches the x-axis at its vertex instead of crossing it.
With real coefficients, any complex root a + bi forces its conjugate a − bi to be a root too, so the two always appear together. This calculator shows both.
The turning point of the parabola, at x = −b/2a. It is a minimum when a > 0 (parabola opens upward) and a maximum when a < 0.