Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 for its roots.
Try an example
4
About this calculator
How This Quadratic Formula Calculator Works
Enter the coefficients a, b, and c of a quadratic equation in the form ax² + bx + c = 0, and the calculator solves for x — including complex roots when the equation has no real solutions.
Worked Example
For x² − 3x − 4 = 0 (a = 1, b = −3, c = −4): the discriminant is 25, giving two real roots, x₁ = 4 and x₂ = −1.
The Formula
x = [−b ± √(b² − 4ac)] ÷ 2a. The expression under the square root, b² − 4ac, is the discriminant — it determines how many real solutions exist before you even compute them.
What the Discriminant Tells You
- Discriminant > 0 — two distinct real roots, where the parabola crosses the x-axis at two points
- Discriminant = 0 — exactly one real root (a repeated root), where the parabola just touches the x-axis at its vertex
- Discriminant < 0 — no real roots; the two solutions are complex conjugates, and the parabola never crosses the x-axis
Worked example
For x² − 3x − 4 = 0, the discriminant is 25 (positive, so two real roots): x = 4 and x = −1.
Frequently asked questions
What does the discriminant tell you?
Its sign predicts the type of roots before you even solve: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.
What are complex roots?
When the discriminant is negative, the square root of a negative number isn't a real number — the roots are expressed using i (the imaginary unit, where i² = −1) instead.
Does this work if a = 0?
No — with a = 0 the x² term disappears and the equation becomes linear (bx + c = 0), which has a different, simpler solution than the quadratic formula.
How do I check a root is correct?
Substitute it back into ax² + bx + c — a correct root makes the expression equal exactly 0.