Quadratic equation solver
Enter the three coefficients. You get the roots, the discriminant that decides how many there are, and the vertex — which is often what the question is really about.
- Second root
- 1.000000
- Nature of the roots
- Two distinct real roots
- Factored form
- 1 (x − 2) (x − 1)
- Discriminant (Δ)
- 1.000000
- Vertex
- (1.5, -0.25)
- Axis of symmetry
- 1.500000
- Parabola
- Opens upward
- y-intercept
- 2.000000
The formula
x = (−b ± √Δ) / 2a with Δ = b² − 4ac. Δ > 0 gives two real roots, Δ = 0 one repeated root, Δ < 0 a conjugate pair. The vertex sits at x = −b/2a.
Worked example
x² − 3x + 2 = 0 has Δ = 9 − 8 = 1, so the roots are (3 ± 1)/2 — that is 2 and 1 — and the vertex is at (1.5, −0.25).
How to use it
- 1Rearrange the equation to equal zero before reading off a, b and c.
- 2Include the sign: in x² − 3x + 2, b is −3.
- 3If a is zero the equation is linear, and it is solved as such rather than rejected.
Questions people ask
What does a negative discriminant mean?
The parabola never crosses the x-axis, so there is no real solution. The two complex roots are still reported, since that is what the algebra gives and it is usually what is being asked for.
Why is the vertex useful?
It is the maximum or minimum of the parabola. In applied problems — trajectory height, profit against price — the vertex is the answer and the roots are incidental.
Last reviewed: 2026-10-20