Quadratic Equation Solver — Roots, Discriminant and Vertex
Both roots, the discriminant and the vertex, from the equation as written.
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Short answer
A quadratic equation can be solved for free in a web browser by typing it either as written — 2x^2 + 5x - 3 = 0 — or as its three coefficients, 2 5 -3. Each line comes back with the discriminant, both roots and the vertex of the parabola. A negative discriminant gives the two complex roots in a ± bi form rather than a refusal, and an equation whose x² coefficient is zero is solved as the linear equation it actually is.
At a glance
| Accepts | One equation, or three coefficients, per line |
|---|---|
| Produces | The discriminant, both roots and the vertex |
| Processing | In your browser |
| Price | Free, no limits |
| Account | Not required |
The quadratic formula is four operations long and still the easiest thing in the world to get wrong, usually in the sign of b or in the 2a underneath. Typing the equation out and reading the answer takes less time than checking your own arithmetic twice — and it scales, because a column of equations is solved in one pass.
The discriminant comes back alongside the roots because it is what is being asked about half the time: whether the parabola crosses the axis twice, touches it once, or misses it entirely. The vertex is there for the other half. It is the maximum or the minimum, which is what a quadratic is usually standing in for in an applied problem.
How to use it
- 1
Type an equation
2x^2 + 5x - 3 = 0 as written, or just the coefficients: 2 5 -3. One equation per line.
- 2
Convert
Each line comes back with its discriminant, both roots and its vertex.
- 3
Read the discriminant
Above zero means two real roots, zero means one repeated root, below zero means two complex ones.
- 4
Copy the result
Copy to the clipboard, or download it as a text file.
Frequently asked questions
What form can the equation be typed in?
Either as an equation in x — 2x^2+5x-3=0, x²-4=0, or 3x^2 = 2x + 1 with terms on both sides — or as the three coefficients a, b and c separated by spaces or commas. Both are how people have an equation to hand.
What happens when the discriminant is negative?
Both complex roots are shown, as a ± bi. There is no real solution in that case, which the negative discriminant has already said; stopping there would just be withholding the rest of the answer.
What if a is zero?
Then it is not a quadratic, and the linear equation that is actually there is solved instead — one root, no discriminant. Coefficients that came out of a spreadsheet column did not choose that case, so refusing it would be unhelpful.
What is the vertex for?
It is the turning point: the minimum of the parabola when a is positive, the maximum when a is negative. In an applied problem that is usually the thing being looked for — the lowest cost, the highest point of a trajectory, the optimal price.