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Quadratic Equation Solver

Q: What does a negative discriminant mean?

The parabola never crosses the x-axis, so there are no real solutions. The two complex conjugate roots are still reported as strings such as '-1+2i, -1-2i'.

Q: Which root is root1?

root1 uses the + sign in the quadratic formula and root2 the − sign, so for a > 0 root1 is the larger root.

Solves ax² + bx + c = 0 with the quadratic formula, reporting the discriminant, both roots (real, or complex conjugates written as strings), the vertex of the parabola and the sum and product of the roots.

When to use

You need the solutions of a second-degree polynomial equation, or the vertex and discriminant of a parabola y = ax² + bx + c.

Do not use when: The coefficient a is 0 (the equation is linear: x = −c / b), or you need roots of x^n = c for other powers (use nth-root).

Formula

Δ = b² − 4ac; x = (−b ± √Δ) / (2a); for Δ < 0: x = −b/(2a) ± i·√(−Δ)/(2|a|); vertex = (−b/(2a), c − b²/(4a))

Inputs

Parameter	Type	Required	Description
`a`	number	yes	Coefficient of x². Must not be 0.
`b`	number	yes	Coefficient of x.
`c`	number	yes	Constant term.

Outputs

Output	Type	Description
`discriminant`	number	Δ = b² − 4ac; positive → two real roots, zero → one repeated root, negative → complex roots.
`nature`	string	Plain-language classification of the roots from the sign of the discriminant.
`root1`	number	(−b + √Δ) / (2a). Only when the roots are real.
`root2`	number	(−b − √Δ) / (2a). Only when the roots are real.
`roots`	string	Both roots as text; complex roots are written like '-1+2i, -1-2i'.
`vertex_x`	number	x-coordinate of the parabola's vertex, −b / (2a) (also the axis of symmetry).
`vertex_y`	number	y-coordinate of the vertex, c − b² / (4a) (the minimum for a > 0, maximum for a < 0).
`sum_of_roots`	number	−b / a (Vieta's formula).
`product_of_roots`	number	c / a (Vieta's formula).

Example

x² − 5x + 6 = 0: {"a":1,"b":-5,"c":6} → {"discriminant":1,"nature":"Two distinct real roots","root1":3,"root2":2,"roots":"3, 2","vertex_x":2.5,"vertex_y":-0.25,"sum_of_roots":5,"product_of_roots":6}

x² + 2x + 5 = 0: {"a":1,"b":2,"c":5} → {"discriminant":-16,"nature":"Two complex conjugate roots (no real roots)","roots":"-1+2i, -1-2i","vertex_x":-1,"vertex_y":4}

GET https://tttkmbb.com/api/v1/calculate/quadratic-equation?a=1&b=-5&c=6

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/quadratic-equation (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/quadratic-equation · Markdown: https://tttkmbb.com/math/quadratic-equation.md · JSON definition: https://tttkmbb.com/math/quadratic-equation.json
MCP: server https://tttkmbb.com/mcp, tool solve_quadratic_equation
OpenAPI operationId: solve_quadratic_equation
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Quadratic formula (reference)
Wolfram MathWorld – Quadratic Equation (reference)

FAQ

What does a negative discriminant mean?

The parabola never crosses the x-axis, so there are no real solutions. The two complex conjugate roots are still reported as strings such as '-1+2i, -1-2i'.

Which root is root1?

root1 uses the + sign in the quadratic formula and root2 the − sign, so for a > 0 root1 is the larger root.

Related calculators

Nth Root Calculator — Solve x^n = c for a single power.
Exponent Calculator — Evaluate powers appearing in polynomial terms.
Slope Calculator — Linear equations from two points.