How to Solve Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. Unlike linear equations, quadratics can have zero, one, or two real solutions — and the discriminant, D = b² − 4ac, tells you which case you have before you compute anything.
Reading the Discriminant
- D > 0: two distinct real roots — the parabola crosses the x-axis twice.
- D = 0: one repeated root — the vertex just touches the x-axis.
- D < 0: no real roots — the solutions are a pair of complex conjugates.
The Quadratic Formula
Plug a, b, and c into x = (−b ± √D) ÷ 2a. The ± is what produces the two roots. Many students lose marks on sign errors with −b, so write each substitution out — exactly what the step-by-step panel above does for you.
Vertex and Graphing
The vertex (−b ÷ 2a) is the parabola's turning point: a minimum when a > 0, a maximum when a < 0. Knowing the vertex plus the roots lets you sketch the entire graph in seconds.
Working with linear equations instead? Try our linear equation solver.