Quadratic Formula Calculator

The Quadratic Formula Calculator solves any quadratic equation of the form ax² + bx + c = 0. Enter the three coefficients a, b, and c, and it instantly finds both roots using the quadratic formula — showing each step clearly.

The quadratic formula is: x = (−b ± √(b² − 4ac)) / 2a

The key to understanding the result is the discriminant (D = b² − 4ac):

• D > 0 — Two distinct real roots. The parabola crosses the x-axis at two points.
• D = 0 — One repeated real root. The parabola just touches the x-axis at one point (vertex on x-axis).
• D < 0 — No real roots. The parabola doesn't cross the x-axis. The roots are complex numbers involving i (√−1).

Quadratic equations appear in physics (projectile motion), economics (profit maximisation), engineering, and geometry. Mastering this formula is one of the most useful skills in mathematics.

Example

Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6).

Step 1: Discriminant D = (−5)² − 4×1×6 = 25 − 24 = 1.

Step 2: Since D = 1 > 0, there are two distinct real roots.

Step 3: x = (5 ± √1) / 2 = (5 ± 1) / 2

x₁ = (5 + 1) / 2 = 3, x₂ = (5 − 1) / 2 = 2

Verification: (x−3)(x−2) = x² − 5x + 6 ✓