Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Finds real or complex roots, the discriminant, and the vertex of the parabola.
x₁
2
x₂
1
Discriminant (b² − 4ac)
1
Nature of roots
Two distinct real roots
Vertex of the parabola
(1.5, -0.25)
Quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant b² − 4ac tells you the kind of roots: positive → two real roots, zero → one repeated root, negative → two complex conjugate roots written as a ± bi.
Frequently Asked Questions
What is the quadratic formula?
For an equation ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. The ± means there are usually two roots: one using the plus sign and one using the minus sign. This calculator computes both for you.
What is the discriminant and what does it tell me?
The discriminant is the part under the square root: b² − 4ac. If it is positive there are two distinct real roots; if it is zero there is one repeated real root; if it is negative there are two complex (non-real) roots that are conjugates of each other.
How do I solve a quadratic equation with complex roots?
When the discriminant is negative, the square root is of a negative number, which produces an imaginary part. The roots take the form (−b ± i√(4ac − b²)) ÷ 2a, written as a ± bi. This calculator detects that case and shows both complex conjugate roots.
Why must coefficient a not be zero?
If a = 0 there is no x² term, so the equation is linear (bx + c = 0), not quadratic, and the quadratic formula doesn't apply. The calculator flags this and asks for a non-zero a.
How do I find the vertex of the parabola?
The vertex of y = ax² + bx + c is at x = −b ÷ 2a, and you find the y-coordinate by plugging that x back into the equation. The vertex is the highest or lowest point of the parabola; the calculator reports it alongside the roots.
Can the quadratic formula solve any quadratic equation?
Yes. Unlike factoring, which only works cleanly for some equations, the quadratic formula solves every quadratic — including those with irrational or complex roots. It's the most reliable method when an equation doesn't factor nicely.