Apply the quadratic formula to any ax² + bx + c, with the discriminant, roots, vertex and factored form.
How the quadratic formula calculator works
The quadratic formula solves every quadratic, but the discriminant tells you what kind of answer to expect before you compute it: positive gives two real roots, zero gives one repeated root, negative gives a complex pair.
The calculator reports all of that plus the vertex, which is the turning point of the parabola and the value you need for maximum and minimum problems.
Worked example
On the values this calculator opens with, the factored form is (2x + 1)(x + 3). Underneath, Discriminant comes out at 25 and Roots at x = -0.5 , x = -3. Change any field and every figure updates as you type.
How to fill it in
- Enter a, b and c from ax² + bx + c = 0.
- Read the discriminant to see what kind of roots to expect.
- Read the roots, vertex and factored form.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a
vertex = (−b/2a, c − b²/4a)
Questions people ask about this
What if a is zero?
Then it is not a quadratic — it reduces to a linear equation, and the calculator solves that instead.
What are complex roots?
When the discriminant is negative the square root is imaginary, giving roots of the form p ± qi that come in conjugate pairs.