The discriminant tells you the answer before you finish
b² − 4ac is the part under the square root, and its sign decides everything. Positive gives two real roots — the curve crosses the axis twice. Zero gives one repeated root, where the curve just touches. Negative means the square root is imaginary and there are no real roots; the parabola floats entirely above or below the axis.
The graph makes this immediate: the shape and the discriminant are saying the same thing two different ways.
Vertex, symmetry and the shortcuts
The vertex sits at x = −b/(2a) — the axis of symmetry, and the midpoint of the two roots whenever real roots exist. It is the minimum when a is positive and the maximum when a is negative, which is what makes quadratics the standard model for optimisation problems.
Two more relations worth carrying: the roots sum to −b/a and multiply to c/a. They give you a quick check on any answer, and they let you reconstruct an equation from roots you already know.
When a = 0
The equation stops being quadratic and becomes linear, with a single root at −c/b. The formula itself breaks down because it divides by 2a. This page detects that and solves the linear equation instead of returning an error or, worse, a silent infinity.
Frequently asked questions
What does the 'i' in the complex roots mean?
It is the imaginary unit, the square root of −1. When the discriminant is negative the roots are complex conjugates — a pair of the form p ± qi. They are perfectly valid solutions, they simply do not appear on a real-number graph.
Is anything uploaded?
No. The roots are worked out locally, with nothing transmitted.
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