Quadratic Equation Solver
Free quadratic equation solver — real and complex roots, discriminant, and vertex of ax² + bx + c = 0, with the formula shown.
Everyday Uses
Algebra homework
Solve any quadratic with the discriminant and vertex shown — learn the why, not just the answer.
Projectile problems
Height-vs-time physics problems are quadratics — find when the ball lands (roots) and its peak (vertex).
Optimization questions
Max profit and min cost problems reduce to finding a parabola's vertex — computed directly here.
Complex roots included
Negative discriminant? Get the full complex answer instead of a dead-end "no solution."
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a solves any equation of the form ax² + bx + c = 0. The ± gives the two roots: one adding the square root term, one subtracting it. It works for every quadratic, including those that don't factor nicely.
What does the discriminant tell me?
The discriminant is b² − 4ac, the expression under the square root. Positive → two distinct real roots (the parabola crosses the x-axis twice). Zero → one repeated root (the parabola touches the axis at its vertex). Negative → two complex conjugate roots (the parabola never touches the x-axis).
What are complex roots and are they valid answers?
When the discriminant is negative, the roots take the form p ± qi, where i = √(−1). They are perfectly valid mathematically and appear as conjugate pairs. In many applied contexts (like lengths and times) they signal "no real solution exists", but in fields like electrical engineering and control theory complex roots carry real physical meaning.
What is the vertex of a parabola?
The turning point, at x = −b/2a. If a > 0 the parabola opens upward and the vertex is the minimum; if a < 0 it opens downward and the vertex is the maximum. The vertex is essential for optimization problems — maximum profit, peak height of a projectile, minimum cost.