quadratic equation calculator

Use this calculator to solve equations of the form ax² + bx + c = 0. Enter your coefficients, hit the button, and instantly see the discriminant and the nature of the roots.

How the Quadratic Formula Works

The quadratic formula x = [-b ± √(b² – 4ac)] / (2a) solves any quadratic equation as long as coefficient a is not zero. The discriminant Δ = b² – 4ac tells us how many real solutions exist. Δ > 0 gives two distinct real solutions, Δ = 0 gives one repeated real solution, and Δ < 0 produces a pair of complex conjugates.

To use the tool, enter the coefficients exactly as they appear in your equation. For example, if your equation is 4x² – 6x + 2 = 0, then a = 4, b = -6, and c = 2. The button computes Δ and the resulting roots automatically.

Example Calculations

Example 1: For 2x² + 3x – 5 = 0, enter a = 2, b = 3, c = -5. The discriminant is 49, so two real roots appear: x₁ = 1 and x₂ = -2.5.

Example 2: For x² + 4x + 8 = 0, set a = 1, b = 4, c = 8. The discriminant is -16, indicating complex solutions: x = -2 ± 2i.

Example 3: For 3x² – 18x + 27 = 0, enter a = 3, b = -18, c = 27. Δ equals 0, so the equation has one repeated root at x = 3.