Quadratic Formula Solver (Step by Step)

Enter the three coefficients of ax² + bx + c = 0 and this solver substitutes them into the quadratic formula one step at a time, the way a teacher would write it on the board. It simplifies the square root (√12 becomes 2√3), reduces the fraction, and gives both the exact roots and their decimal values, so you can check your working as well as the final answer.

How to use this tool

1. Rewrite your equation as ax² + bx + c = 0, moving every term to one side first.
2. Enter a, b and c, including any minus signs, for example b = −4.
3. Read the exact solutions and the decimal roots.
4. Follow the step table to see the substitution, the discriminant and the simplification.

Formula

x = (−b ± √(b² − 4ac)) / (2a)

The ± gives two answers, one with + and one with −. The expression under the root, D = b² − 4ac, is the discriminant. A perfect-square D gives rational roots, any other positive D gives irrational roots written with √, D = 0 gives a single repeated root, and a negative D gives two complex roots written with i.

Example

Solve x² − 4x + 1 = 0, so a = 1, b = −4 and c = 1. The discriminant is (−4)² − 4 × 1 × 1 = 16 − 4 = 12. Simplify √12 = √(4 × 3) = 2√3, so x = (4 ± 2√3) / 2. Dividing every term by 2 gives x = 2 ± √3, about 0.267949 and 3.732051. Check: the roots add up to 4 (= −b/a) and multiply to 1 (= c/a).