Quadratic Equation Solver

The vertex is the highest or lowest point of the parabola y = ax² + bx + c. Its x-coordinate is −b / (2a) — the axis of symmetry. The y-coordinate is the value of the function at that x. If a > 0 the vertex is a minimum; if a < 0 it's a maximum.

Vieta's relations connect the roots to the coefficients without solving. For ax² + bx + c = 0: sum of roots = −b/a, product of roots = c/a. Useful for quick sanity checks: if a=1, b=−5, c=6, the roots must sum to 5 and multiply to 6 (they're 2 and 3).

When the discriminant is non-negative, you can write the equation as a × (x − r₁) × (x − r₂) = 0, where r₁ and r₂ are the roots. This form makes the roots obvious by inspection. Customize mode displays it when real roots exist.

Complex roots take the form p ± qi, where p is the real part and qi is the imaginary part. For ax² + bx + c = 0 with negative discriminant: p = −b / (2a), q = √(4ac − b²) / (2a). The calculator displays both parts separately.

If a = 0, the x² term vanishes and the equation becomes bx + c = 0 — a linear equation with at most one solution. The quadratic formula divides by 2a, so a = 0 makes the formula undefined.

The y-intercept is the value of f(x) at x = 0 — i.e. y = c. It's where the parabola crosses the y-axis. Combined with the vertex and any real roots, it's enough to sketch the parabola accurately by hand.