Quadratic Equation Solver
Plug in a, b, c and get the roots, the discriminant, the vertex, the axis of symmetry, and a graph of the parabola.
Solve a·x² + b·x + c = 0 with full working — discriminant, vertex, axis of symmetry and graph.
The quadratic formula in one line
For any equation of the form ax² + bx + c = 0 (with a ≠ 0), the two roots are given by:
x = (−b ± √(b² − 4ac)) / (2a)
The expression inside the square root is the discriminant. Its sign tells you what kind of roots to expect before you even compute them.
Vertex form and axis of symmetry
Every parabola is symmetric about a vertical line called its axis of symmetry. For ax² + bx + c the axis is at x = −b / (2a), and the vertex — the lowest or highest point — sits exactly on that line. Using the vertex form a(x − h)² + k makes graphing easier, but the calculator handles both forms transparently.
Sum and product of roots
By Vieta’s formulas, the two roots r₁ and r₂ of ax² + bx + c = 0 satisfy r₁ + r₂ = −b/a and r₁ · r₂ = c/a. That is a quick sanity check on any quadratic solution — add the two roots, and you should get −b/a back.
Worked example
Take 2x² − 5x + 2 = 0. Here a = 2, b = −5, c = 2. The discriminant is 25 − 16 = 9, which is positive, so there are two real roots. Applying the formula gives x = (5 ± 3) / 4, that is x = 2 or x = 1/2. The product 2 × ½ is 1, which is c/a, and the sum is 2.5, which is −b/a — our sanity checks pass.
FAQ
What does the discriminant tell me?
The discriminant D = b² − 4ac determines the nature of the roots. D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives a pair of complex-conjugate roots.
Why does the parabola sometimes open downwards?
When the leading coefficient a is negative, the parabola opens downwards and the vertex is the maximum value of f(x).
What if a is zero?
Then the equation is linear, not quadratic. The solver will refuse the input and ask you to use the general equation solver instead.