Quadratic Formula Calculator
Enter the coefficients a, b and c to solve ax² + bx + c = 0. Shows the discriminant and the real or complex roots.
Reviewed by the WorldCalcs team · Methodology · Last reviewed: June 2026
Result
Discriminant (D): 1
Type: two real
x₁ = 3
x₂ = 2
What is the quadratic formula?
A quadratic equation has the form ax^2 + bx + c = 0, where a is not zero. The quadratic formula solves it: x equals minus b, plus or minus the square root of (b^2 minus 4ac), all divided by 2a. It always works, whatever the coefficients, and gives the points where the parabola crosses the x-axis. This calculator returns the roots and the discriminant for any a, b and c.
How it's calculated
The key part is the discriminant, D = b^2 - 4ac, which tells you what kind of roots you get. If D is positive there are two different real roots; if D is zero there is one repeated (double) root; if D is negative there are two complex roots that form a conjugate pair. Put a, b and c into the formula, work out the discriminant, take its square root, and apply the plus and the minus.
Example
Solve x^2 - 5x + 6 = 0, so a = 1, b = -5, c = 6. The discriminant is (-5)^2 - 4(1)(6) = 25 - 24 = 1, which is positive, so there are two real roots. The square root of 1 is 1, giving x = (5 + 1) divided by 2 = 3 and x = (5 - 1) divided by 2 = 2. The solutions are 3 and 2.
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Results are estimates and may contain errors — for general information only, not professional advice. Always verify before relying on them. Disclaimer
How to use
Enter the three coefficients a, b and c of your quadratic equation. The calculator computes the discriminant and the roots.
If the discriminant is positive you get two real roots; if zero, one repeated root; if negative, a conjugate pair of complex roots. If a is zero the calculator flags that the equation is linear, not quadratic.