Quadratic Equation Solver

How to solve a quadratic equation

A quadratic equation is any equation you can write as ax² + bx + c = 0, where a ≠ 0. The reliable way to solve one is the quadratic formula, which works for every quadratic — even when factoring is impossible and the roots are complex.

1. Identify a, b, and c. For x² − 5x + 6 = 0, a = 1, b = −5, c = 6. Watch the signs — b is −5 here, not 5.
2. Compute the discriminant Δ = b² − 4ac. It tells you what kind of roots to expect: Δ > 0 means two distinct real roots, Δ = 0 means one repeated root, Δ < 0 means two complex roots.
3. Plug into the quadratic formula: x = (−b ± √Δ) / 2a. Substitute your a, b, c and the discriminant you just found.
4. Simplify. Evaluate the ± branch separately to get both roots, and reduce any fractions or radicals.

How to find the vertex of a parabola

To find the vertex of a parabola, compute h = −b / 2a and then k = f(h) — plug h back into the equation. For x² − 4x + 3, h = 2 and k = −1, so the vertex is (2, −1). The vertical line x = h is the axis of symmetry, and c is the y-intercept (where the parabola crosses the y-axis).

Reading the discriminant

Think of the discriminant as the equation's fortune-teller. A positive discriminant (like Δ = 1 for x² − 5x + 6) means the parabola crosses the x-axis twice — roots 2 and 3. Δ = 0 means it just touches the axis (x² − 4x + 4 touches at x = 2). A negative discriminant (Δ = −4 for x² + 1) means the parabola never touches the x-axis, so the roots are complex: ±i.