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.

The difference is that the quadratic formula is a universal method that solves any quadratic equation, whereas factoring is a shortcut that only works easily when the roots are integers. If a quadratic expression cannot be factored using simple numbers, the quadratic formula must be used to find the solutions.

The variables a, b, and c represent the numerical coefficients of the terms in a standard quadratic equation. Specifically, a is the coefficient of the squared term (x²), b is the coefficient of the linear term (x), and c is the constant number.

You use completing the square primarily when solving quadratic equations that are not easily factorable or when rewriting a standard form quadratic function into vertex form to find its vertex. While the quadratic formula is generally faster for finding roots, completing the square is essential for understanding the algebraic origin of the formula itself.

No, a quadratic equation always has exactly two solutions, though they may be real and distinct, real and identical, or complex conjugate numbers. In real-world problems where only real-number solutions are physically possible, a negative discriminant is often interpreted as having no real solutions.

The term −b / 2a in the quadratic formula gives the x-coordinate of the parabola's vertex (which is also the axis of symmetry). The roots of the equation are spaced symmetrically at a distance of √(b² − 4ac) / 2a to the left and right of this vertex line.