Quadratic Formula Calculator

The quadratic formula solves any equation in the form ax² + bx + c = 0. The two roots are x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant; it tells you whether the roots are real and distinct, real and repeated, or complex.

Identify the three coefficients a, b, and c from your equation. Compute the discriminant b² − 4ac. If it is positive, take its square root and use the formula to find two real roots. If it is zero, the formula gives one repeated root. If it is negative, the roots are a complex conjugate pair.

The discriminant (D = b² − 4ac) describes how many real roots the equation has. D > 0 means two distinct real roots, D = 0 means one repeated real root, and D < 0 means no real roots (the two roots are complex conjugates).

The square root of a negative number is not a real number, so the two roots are complex. They come in a conjugate pair: one is real_part + imaginary_part·i and the other is real_part − imaginary_part·i. The real part is −b / 2a and the imaginary part is √(−D) / 2a.

Yes, but only when the discriminant is exactly zero. In that case, the formula collapses to a single value, x = −b / 2a, which is sometimes called a repeated or double root because it counts twice in the algebra.

If a = 0, the x² term disappears and the equation becomes bx + c = 0, which is linear, not quadratic. The quadratic formula divides by 2a, so a = 0 would also mean dividing by zero. The calculator flags this case and asks you to enter a nonzero a.

Yes. The inputs accept decimals, fractions written as decimals, and negative numbers. Very large or unusual values are handled gracefully; the calculator rounds the displayed roots for readability while computing internally at full floating-point precision.