Quadratic Formula Calculator
Reading the discriminant
The quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a succeeds for every quadratic, and the piece under the square root — the discriminant b² − 4ac — predicts the answer’s shape before you compute it. A positive discriminant gives two different real roots, exactly zero gives one root repeated, and a negative discriminant gives a complex-conjugate pair with no real solutions, since square roots of negatives are not real. The formula works through completing the square: divide through by a, shift the constant, and solve for the squared term.
Frequently Asked Questions
What does the vertex tell me?
It is the turning point of the parabola y = ax² + bx + c, sitting on the axis of symmetry x = −b ÷ 2a. For a > 0 the parabola opens upward and the vertex is the minimum; for a < 0 it is the maximum.
Why do complex roots come in pairs?
A quadratic with real coefficients cannot have exactly one complex root — its conjugate must also be a root, so they appear as p + qi and p − qi sharing the same real part and opposite imaginary parts.