NumberTally Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 in seconds. Enter the coefficients a, b, and c to get the discriminant and both roots — real, repeated, or complex — with every step of the work shown.
Advanced options
The discriminant decides everything
Before computing roots, look at D = b² − 4ac. Positive → two distinct real roots (the parabola crosses the x-axis twice). Zero → one repeated root (it just touches). Negative → two complex conjugate roots (it never touches). The comparison box below shows one example of each.
Vertex and factored form
The vertex at x = −b/2a is the parabola's turning point — its minimum if a > 0, maximum if a < 0. When roots are real, the equation factors as a(x − r₁)(x − r₂), which the calculator shows alongside the graph.
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b²−4ac)) / 2a. It solves any quadratic ax²+bx+c = 0, derived by completing the square.
What is the discriminant and what does it tell me?
D = b²−4ac. D > 0: two real roots; D = 0: one repeated root; D < 0: two complex roots.
How many solutions does a quadratic equation have?
Always two counting multiplicity — real or complex. A "single" solution is really a repeated root.
What happens if a equals zero?
It stops being quadratic: bx + c = 0 is linear, with the single root x = −c/b. The calculator handles this case.
What does a negative discriminant mean?
No real roots — the parabola never crosses the x-axis. The roots are complex conjugates, e.g. 1 ± 2i.