Cubic Real Root Finder
Find a real root of ax³ + bx² + cx + d = 0 numerically, then factor out the remaining quadratic for any other real roots.
How it works
Solve ax³ + bx² + cx + d = 0 · find one real root by Newton's method · deflate to a x² + (b+ar)x + (c+(b+ar)r) = 0
Every cubic with real coefficients has at least one real root, because its graph runs from −∞ to +∞. This tool finds that root numerically with Newton's method, then divides it out (synthetic deflation) to leave a quadratic whose discriminant reveals whether the other two roots are real or a complex pair. It is a simple, robust approach rather than the full Cardano formula.
Worked example
For x³ − 6x² + 11x − 6 = 0, Newton's method converges to x = 1. Deflating gives x² − 5x + 6 = 0, whose roots are x = 2 and x = 3. So the cubic has three real roots: 1, 2 and 3.
Sources: Cubic equation — Wikipedia · Newton's method — Wikipedia
Frequently asked questions
Why use Newton's method instead of an exact formula?
The exact Cardano formula for cubics involves nested cube roots and complex arithmetic even when all roots are real (the "casus irreducibilis"), which is error-prone to display. Newton's method converges quickly to a real root, and deflating to a quadratic then gives the rest exactly — a simpler and numerically stable path.
How many real roots can a cubic have?
A cubic always has one or three real roots (counting a repeated root as one value). It can never have exactly two distinct real roots and one leftover: complex roots always come in conjugate pairs, so the count of non-real roots is 0 or 2.
What if my leading coefficient a is zero?
Then the x³ term vanishes and the equation is quadratic (or lower), not cubic. The finder reports this so you can switch to the quadratic solver, which handles ax² + bx + c = 0 directly.
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This calculator is for educational purposes. Double-check important results before acting on them.