Find one root, then the rest. Any cubic, solved.

The general strategy for ax³ + bx² + cx + d = 0 is: find one root first, then reduce to a quadratic. AskSia uses the rational root theorem to list candidate rational roots: any rational root p/q must have p as a factor of d (the constant term) and q as a factor of a (the leading coefficient). AskSia tests each candidate by substitution or synthetic division. Once a root r is found, dividing by (x minus r) via synthetic division gives a quadratic factor, which AskSia solves with the quadratic formula to get the remaining two roots. The full process produces all three roots in exact and decimal form.

The rational root theorem says that for a polynomial with integer coefficients, any rational root p/q (in lowest terms) must satisfy: p is a factor of the constant term, and q is a factor of the leading coefficient. For example, for 2x³ minus 3x² minus 8x + 12 = 0, the constant is 12 (factors: ±1, ±2, ±3, ±4, ±6, ±12) and the leading coefficient is 2 (factors: ±1, ±2). Candidate rational roots are ±1, ±1/2, ±2, ±3, ±3/2, ±4, ±6, ±12. AskSia tests each by substituting into the cubic and finds which ones make it zero.

Yes, but always in conjugate pairs (when coefficients are real). A cubic with real coefficients has either three real roots (which may include repeats), or one real root and a pair of complex conjugate roots. Cubic equations always have at least one real root because cubics extend from negative infinity to positive infinity, so they must cross the x-axis somewhere. AskSia identifies the case (after finding the first root) and reports the remaining two roots as either two reals or one complex conjugate pair in a + bi form.

If the rational root theorem produces no working candidates, the cubic has only irrational or complex roots. AskSia falls back to either Cardano's method (which gives exact closed-form expressions involving nested radicals) or numerical methods (Newton's method, bisection) to find an approximate first root, then proceeds with the standard synthetic division and quadratic-finish. For most coursework problems through Algebra 2 and College Algebra, rational roots exist by design, so the rational root theorem succeeds.

AskSia hits 98% accuracy on standard high school and college coursework, measurably higher than ChatGPT, Photomath, and Symbolab on the same problem sets. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.

Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.

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