{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"quadratic-equation"},"result":{"entity_type":"calculator","id":"quadratic-equation","calculator_id":"quadratic-equation","canonical_url":"https://tttkmbb.com/math/quadratic-equation","name":"Quadratic Equation Solver","title":"Quadratic Equation Solver – Roots, Discriminant and Vertex of ax² + bx + c = 0","category":"math","category_name":"Math","tool_name":"solve_quadratic_equation","featured_mcp_tool":true,"description":"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.","use_when":"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).","inputs":[{"name":"a","label":"a (x² coefficient)","type":"number","required":true,"description":"Coefficient of x². 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Inputs supplied by the caller determine the output."},"examples":[{"name":"x² − 5x + 6 = 0","inputs":{"a":1,"b":-5,"c":6},"expected":{"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},"url":"https://tttkmbb.com/api/v1/calculate/quadratic-equation?a=1&b=-5&c=6"},{"name":"x² + 2x + 5 = 0","inputs":{"a":1,"b":2,"c":5},"expected":{"discriminant":-16,"nature":"Two complex conjugate roots (no real roots)","roots":"-1+2i, -1-2i","vertex_x":-1,"vertex_y":4},"url":"https://tttkmbb.com/api/v1/calculate/quadratic-equation?a=1&b=2&c=5"}],"faq":[{"q":"What does a negative discriminant mean?","a":"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?","a":"root1 uses the + sign in the quadratic formula and root2 the − sign, so for a > 0 root1 is the larger root."}],"tags":["quadratic equation","quadratic formula","discriminant","roots","parabola vertex","solve ax2+bx+c"],"related":[{"calculator_id":"nth-root","reason":"Solve x^n = c for a single power."},{"calculator_id":"exponent","reason":"Evaluate powers appearing in polynomial terms."},{"calculator_id":"slope","reason":"Linear equations from two points."}],"links":{"html":"https://tttkmbb.com/math/quadratic-equation","markdown":"https://tttkmbb.com/math/quadratic-equation.md","json":"https://tttkmbb.com/math/quadratic-equation.json","api":"https://tttkmbb.com/api/v1/calculate/quadratic-equation","schema":"https://tttkmbb.com/api/v1/calculators/quadratic-equation","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-23"},"timestamp":"2026-09-23T23:27:56Z"}