Quadratic Formula Calculator

x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any quadratic equation, and it comes from completing the square on the general form.

b² − 4ac decides the nature of the roots before you calculate them. Positive gives two real roots, zero gives one repeated root at the vertex, negative gives a complex pair.

The parabola never crosses the x-axis, so there are no real roots. The complex roots are still meaningful, and the vertex and shape of the curve are perfectly real — which is why they are shown.

The x-coordinate is −b ÷ 2a, which is also the axis of symmetry. Substitute it back to get the y-coordinate. The vertex is the minimum if a is positive and the maximum if negative.

The roots sum to −b/a and multiply to c/a. They are the fastest check on a solution, and they work even when the roots are complex.

When the roots are rational, yes, and it is quicker. The formula always works, which is why it is the fallback when factoring does not present itself.

By completing the square on ax² + bx + c = 0. Divide by a, move c across, add the square of half the x-coefficient to both sides, and solve — the formula falls straight out.

Rewriting a quadratic as a perfect square plus a constant, which makes the vertex readable directly. It is also how the formula is derived, so the two are the same method at different levels of abstraction.

When the roots are rational and the factors are visible. It is quicker when it works, and the formula always works — which is why it is the fallback.

In engineering and physics they describe oscillation. The imaginary part corresponds to a frequency and the real part to growth or decay, which is central to signal processing and control theory.

Yes, when the discriminant is zero — the parabola touches the x-axis at its vertex rather than crossing. It is usually described as a repeated root rather than a single one.

Both have general formulas, though they are far more complicated. Above degree four no general formula exists — the Abel–Ruffini theorem proved that permanently rather than as a limit of technique.