Quadratic Formula Calculator. Enter the coefficients a, b, and c from your quadratic equation (ax² + bx + c = 0) and get back both roots — x₁ and x₂ — along with the discriminant and the nature of roots. Works for real and complex solutions. Also try the use the Diamond Problem Calculator.
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic equation of the form ax² + bx + c = 0 for x, given that a ≠ 0. See also our find Factored Form with Factoring Calculator.
The discriminant is the expression b² − 4ac. If it's greater than 0, there are two distinct real roots. If it equals 0, there is exactly one real root (a repeated root). If it's less than 0, the roots are complex (imaginary).
No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0 to be valid.
When the discriminant (b² − 4ac) is negative, the square root of a negative number is involved, producing complex (imaginary) roots. These are expressed in the form p ± qi, where i = √(−1). You might also find our find Slope (m) with Slope Calculator useful.
When the discriminant equals exactly 0, both roots are identical. This is called a repeated or double root, and the parabola touches the x-axis at exactly one point.
For an equation written as ax² + bx + c = 0, a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, in 3x² − 5x + 2 = 0, a = 3, b = −5, and c = 2.
Yes. You can enter any real number for a, b, and c — including negative values and decimals. The calculator correctly applies the quadratic formula for all valid inputs.
For a quadratic equation ax² + bx + c = 0, the sum of the roots equals −b/a and the product of the roots equals c/a. These are useful properties for verifying your solutions. Check out our Math Solver (Step-by-Step) as well.