Quadratic Equation Solver
Enter the coefficients a, b and c of ax² + bx + c = 0 and get the discriminant, the real or complex roots, and every step of the quadratic formula.
Non-exact values are rounded to six decimal places.
What the discriminant tells you before you solve anything
The discriminant Δ = b² − 4ac settles the whole story before you compute a single root. When Δ > 0 there are two distinct real roots, and the parabola y = ax² + bx + c crosses the x-axis at two points. When Δ = 0 there is one repeated root, and the parabola just touches the x-axis at its vertex. When Δ < 0 there are no real roots — the parabola never reaches the x-axis — but there are two complex conjugate roots, which this solver works out for you anyway.
The most common mistakes (and how to dodge them)
Mistake number one is the sign of b in the formula x = (−b ± √Δ)/(2a): if b is already negative, −b flips to positive — with b = −3 you get −b = 3, not −3. Mistake number two is forgetting to rewrite the equation in standard form before reading off the coefficients: for 3x² = 5x − 2 you first need 3x² − 5x + 2 = 0. Finally, if a = 0 the equation isn't quadratic at all — it's linear. This tool notices that on its own and solves it as bx + c = 0.