{"uid":"cap_DrVxC7VwGGkoPgAfdmTJy","slug":"cubic-equation-solver-ax-bx-cx-d-0-329a2e23","name":"Cubic Equation Solver (ax³ + bx² + cx + d = 0)","description":"Solve ax^3 + bx^2 + cx + d = 0: all three roots (real and complex) by Cardano and the trigonometric method.","url":"https://math-geometry.x402tookit.com/cubic-solve?utm_source=zero.xyz","method":"GET","headers":{},"bodySchema":{"type":"object","$schema":"https://json-schema.org/draft/2020-12/schema","required":["input"],"properties":{"input":{"type":"object","required":["type","method"],"properties":{"type":{"type":"string","const":"http"},"method":{"enum":["GET"],"type":"string"},"queryParams":{"type":"object","required":["a","b","c","d"],"properties":{"a":{"type":"number","description":"Coefficient of x^3 (not 0)"},"b":{"type":"number","description":"Coefficient of x^2"},"c":{"type":"number","description":"Coefficient of x"},"d":{"type":"number","description":"Constant term"}}}},"additionalProperties":false},"output":{"type":"object","required":["type"],"properties":{"type":{"type":"string"},"example":{"type":"object","properties":{"a":{"type":["null","number"]},"b":{"type":["null","number"]},"c":{"type":["null","number"]},"d":{"type":["null","number"]},"asOf":{"type":["null","string"]},"roots":{"type":["array","null"],"items":{"type":"string"}},"source":{"type":["null","string"]},"reasons":{"type":["array","null"],"items":{"type":"string"}},"summary":{"type":["null","string"]},"verdict":{"type":["null","string"]},"degraded":{"type":["boolean","null"]},"rootType":{"type":["null","string"]},"depressed":{"type":["null","object"],"properties":{"p":{"type":["null","number"]},"q":{"type":["null","number"]},"discriminant":{"type":["null","number"]}}},"realRoots":{"type":["array","null"],"items":{"type":"number"}},"disclaimer":{"type":["null","string"]},"notChecked":{"type":["array","null"],"items":{"type":"string"}},"complexRoots":{"type":["array","null"]}}}}}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.02","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.02/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.02","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.02","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_tj4qSlcxcaL09K8Y8HFur","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.02","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Solves a cubic equation given coefficients a, b, c, d and returns all three roots (real and complex) using Cardano's formula and the trigonometric method.","exampleAgentPrompt":"Solve the cubic equation 2x³ - 3x² + x - 5 = 0 and give me all three roots, including any complex ones.","exampleUseCases":[{"title":"Engineering cubic root finding","prompt":"I need all three roots of x³ - 6x² + 11x - 6 = 0 — can you solve it and tell me if they're all real or if any are complex?"},{"title":"Physics resonance frequency calculation","prompt":"Solve 3x³ + 2x² - x + 4 = 0 for me and show the discriminant and the depressed cubic form too."},{"title":"Student homework check","prompt":"What are the roots of the cubic equation x³ + 0x² - 3x + 2 = 0? I need all three roots and whether they're real or complex."}],"resultDescription":"Returns all three roots of the cubic as strings (supporting complex notation), a rootType classification (e.g. three real, one real + two complex), the discriminant, the depressed cubic coefficients p and q, separate arrays of realRoots and complexRoots, and a human-readable summary. May also include a disclaimer about numerical precision.","failureModes":["Coefficient a = 0 (not a cubic): rejected or falls back to quadratic","Malformed or missing query parameters a, b, c, d: 400 error","Very large or very small coefficients causing numerical overflow","Network or payment failure returning 402 or 5xx"],"whenToPreferThis":"Use this endpoint when you need an analytic, symbolic-style solution for a cubic polynomial (degree exactly 3), especially when you need all three roots including complex ones and want intermediate results like the discriminant and depressed cubic. Prefer it over numerical root-finders when exact Cardano/trigonometric closed-form methodology is required or when root type classification matters.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-10-30T06:37:48.052Z","isFirstParty":false,"canonicalSlug":"cubic-equation-solver-ax-bx-cx-d-0-329a2e23"}