Quadratic Formula Calculator
Solve any equation of the form ax² + bx + c = 0. See the parabola, the roots, the vertex, and every step of the working — with support for fractions and decimals.
What is a quadratic equation?
A quadratic equation is any polynomial equation of degree two — the highest power of the unknown is 2. In standard form it looks like ax² + bx + c = 0, where a ≠ 0. Graphed, y = ax² + bx + c is a parabola: it opens upward when a > 0 and downward when a < 0. This calculator uses the quadratic formula because it works for every case — integer, irrational, or complex roots.
Quadratic formula, piece by piece
Each card explains one part of what happens when you press Solve — from the formula itself and its derivation, through the discriminant, to the three shapes the answer can take.
The quadratic formula
The formula returns both roots at once. The ± gives the two solutions; the expression under the square root — the discriminant — controls what kind of roots you get.
The discriminant tells you the shape of the answer
Before you even solve, D = b² − 4ac tells you whether the parabola crosses the x-axis twice, touches it once, or misses it entirely.
Where the formula comes from
The formula is what you get by completing the square on the general quadratic. Each line below is one algebraic step from ax² + bx + c = 0 to x = (−b ± √D)/2a.
Complex roots when D < 0
When the discriminant is negative the square root is imaginary and the two roots form a complex conjugate pair a ± bi. The parabola never touches the x-axis, but the algebra still returns two valid roots.
Features of this calculator
- Solves ax² + bx + c = 0 with real or complex roots
- Accepts fractional and decimal coefficients (e.g. 1/4, −0.75)
- Live SVG parabola with labeled roots, vertex, and axis of symmetry
- Full step-by-step working — substitution, discriminant, simplification
- Explains what the discriminant tells you about the roots
Frequently asked questions
+What is the quadratic formula?
The quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), gives the exact solutions to any equation of the form ax² + bx + c = 0 as long as a is not zero.
+What does the discriminant mean?
The discriminant is b² − 4ac. If it is positive, the equation has two distinct real roots; if it is zero, one repeated real root; if it is negative, two complex conjugate roots and no real solution.
+Can the quadratic formula solve any equation?
It solves every quadratic — any equation that fits ax² + bx + c = 0 with a ≠ 0. It does not solve higher-degree polynomials (cubic, quartic, and beyond) directly.
+Can a, b, or c be fractions or decimals?
Yes. This calculator accepts fractions such as 1/4 or -3/2 and decimals such as 0.5 for each coefficient.