Quadratic equation solver

Roots

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ax² + bx + c = 0. The discriminant b² − 4ac decides the shape of the answer: positive gives two real roots, zero gives one repeated real root, and negative gives two complex roots — shown as p ± qi rather than reported as "no solution".

What it solves

Given a, b and c in ax² + bx + c = 0, this finds every x that makes the equation true, using the quadratic formula x = (-b ± sqrt(b² - 4ac)) / 2a.

The discriminant decides the shape of the answer

b² - 4ac is worked out first because its sign tells you what kind of roots to expect before the formula runs. Positive gives two distinct real roots. Zero gives one repeated real root - the parabola just touches the x-axis rather than crossing it. Negative gives no real root at all; the parabola never reaches the axis.

Complex roots, not an error

Most calculators stop at "no real solution". This one continues into complex numbers, because a negative discriminant does not mean there is no answer - it means the answer is not on the real number line. The result is shown as p ± qi, worked out by taking the square root of the discriminant's magnitude and dividing by 2a.

a = 0 is refused, not silently ignored

If a is zero the equation is linear, not quadratic, and dividing by 2a would divide by zero. Rather than crash or quietly solve a different equation than the one entered, this says so and asks for a non-zero a.

Vertex and axis of symmetry

Alongside the roots, the vertex ( -b/2a, f(-b/2a) ) and the axis of symmetry x = -b/2a are shown - the turning point of the parabola and the vertical line it is mirrored around, useful for sketching the curve without plotting every point.