Quadratic equation solver
Enter a, b and c to solve ax² + bx + c = 0.
x₁
x₁: 2
- x₂
- 1
- Exact form
- x = 2, x = 1
- Discriminant (b² − 4ac)
- 1
- Vertex x
- 1.5
- Vertex y
- -0.25
Step by step
- Δ = b² − 4ac = 1
The discriminant is positive: two different real roots.
How it works
The quadratic formula gives both roots at once. The discriminant b² − 4ac decides their type: positive gives two real roots, zero gives one repeated root, negative gives a complex pair.
To avoid losing digits when b² is much larger than 4ac, the calculator uses the numerically stable form of the formula. With whole-number coefficients it also shows the exact answer as a simplified surd.
The vertex is the turning point of the parabola y = ax² + bx + c.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a Δ = b² − 4ac vertex = (−b ÷ 2a, c − b² ÷ 4a)
Worked example
For x² − 3x + 2 = 0: Δ = 9 − 8 = 1, so x = (3 ± 1) ÷ 2, giving x = 2 and x = 1. The vertex is at (1.5, −0.25).
Questions people ask
What if a is zero?
Then the equation is linear, bx + c = 0, with the single solution x = −c ÷ b.
What are complex roots?
When the discriminant is negative the parabola never crosses the x-axis. The roots are then p ± qi, where i² = −1.
Can I factorise instead?
If the exact roots are fractions, yes: roots 2 and 1 mean x² − 3x + 2 = (x − 2)(x − 1).