Quadratic Equation Solver
Enter the coefficients of the equation ax² + bx + c = 0 — you will get the roots, discriminant, vertex and the parabola's graph instantly.
Equation: x² − 5x + 6 = 0
Graph: the parabola with its vertex (green point) and real roots (red points) is shown.
How it works
The quadratic formula is: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant (D = b² − 4ac) tells the type of the roots: if D > 0 there are two different real roots, if D = 0 there is one repeated root, and if D < 0 there are complex roots (p ± qi).
- Example 1: x² − 5x + 6 = 0 — D = 25 − 24 = 1, roots x = 3 and x = 2, vertex (2.5, −0.25).
- Example 2: x² + 2x + 5 = 0 — D = 4 − 20 = −16, complex roots −1 ± 2i (the parabola does not touch the x-axis).
Note: if a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0), and the calculator will answer accordingly.
How to use
- Enter the coefficients a, b and c of your quadratic equation.
- The results update live: roots, discriminant, vertex, axis of symmetry and y-intercept.
- Read the parabola graph to see where it crosses the axes.
Frequently Asked Questions (FAQ)
What does the Quadratic Equation Solver show?
For ax squared + bx + c = 0 it shows the roots (real or complex), the discriminant, the vertex, the axis of symmetry, the y-intercept and a graph of the parabola.
What happens if I set a to 0?
Then the equation is no longer quadratic; it becomes linear (bx + c = 0), and the calculator answers accordingly.
Can it handle complex roots?
Yes. When the discriminant is negative it shows the complex roots in the form p plus or minus qi.
Is the Quadratic Equation Solver free?
Yes. It is free, needs no signup, and runs entirely in your browser.