Quadratic Equation Calculator
Solve ax² + bx + c = 0 with steps
Two distinct rational roots
Working
- x² − 5x + 6 = 0
- D = b² − 4ac = (−5)² − 4 × 1 × 6 = 1
- x = (−b ± √D) ÷ 2a = (5 ± √1) ÷ 2
- x₁ = 3, x₂ = 2
- 1Write it as ax² + bx + c = 0Move everything to one side
- 2Enter a, b and cUse negatives where needed
- 3Press SolveRoots, vertex and working
What is the quadratic formula?
Every quadratic equation can be solved with the quadratic formula. The discriminant D = b² − 4ac decides what kind of roots it has.
x = (−b ± √(b² − 4ac)) ÷ 2a
D > 0: two real roots · D = 0: one repeated root · D < 0: two complex roots
Sum of roots = −b/a · Product of roots = c/a
Nature of roots
When D is a perfect square and a, b, c are whole numbers, the roots are rational and the equation can be factorised. When D is positive but not a perfect square, the roots are irrational, like (5 ± √13)/2.
The vertex
The graph of y = ax² + bx + c is a parabola. Its turning point (vertex) is at x = −b/2a. It opens upward when a > 0, so the vertex is the minimum, and downward when a < 0.
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Frequently asked questions
How do I solve x² − 5x + 6 = 0?
D = 25 − 24 = 1, so x = (5 ± 1)/2, giving x = 3 and x = 2. It factors as (x − 3)(x − 2).
What if the discriminant is negative?
There are no real roots; the parabola doesn't cross the x-axis. The roots are complex: x² + 2x + 5 = 0 gives x = −1 ± 2i.
What if a is 0?
Then it's not a quadratic but a linear equation bx + c = 0, with the single answer x = −c/b.
Last reviewed: October 2026