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
Roots (values of x)
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Discriminant (b² − 4ac)
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Vertex (h, k)
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Axis of Symmetry
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y-intercept
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Parabola opens
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Sum / Product of roots
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.