Quadratic Equation Solver

Suppose you put the default values into Quadratic Equation Solver:

Plug those into the formula x = (−b ± √(b² − 4ac)) / 2a and the result is:

1. Compute the discriminant D = b² − 4ac: with a = 1, b = −3, c = 2, D = 9 − 8 = 1.
2. Take its square root: √1 = 1 (if D is negative, the roots are complex).
3. Apply x = (−b ± √D) / 2a: (3 + 1) ÷ 2 = 2 and (3 − 1) ÷ 2 = 1.
4. Check by substitution: 2² − 3·2 + 2 = 0 and 1² − 3·1 + 2 = 0.
5. For the vertex, use x = −b/2a = 1.5 and y = c − b²/4a = −0.25.

The solver applies the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, the closed-form solution obtained by completing the square on ax² + bx + c = 0. It first evaluates the discriminant D = b² − 4ac, which decides the character of the roots: D > 0 gives two distinct real roots, D = 0 gives one repeated real root at x = −b/2a, and D < 0 gives a complex-conjugate pair, which the calculator writes out as x = re ± im·i rather than reporting no solution. It also reports the parabola's vertex — x = −b/2a and y = c − b²/4a — the turning point on the axis of symmetry midway between real roots. The formula requires a ≠ 0; with a = 0 the equation is linear, not quadratic.

Last reviewed July 2, 2026 · Editorial policy