Mathematical foundation
A cubic has exactly three complex roots when multiplicity is counted. Real coefficients guarantee that nonreal roots occur in conjugate pairs. Cubic Equation Solver also connects with check residuals.
Reading three roots correctly starts with the mathematical structure described here.
Problems this can answer
Cubic equations appear in volume relationships, intersections, optimization conditions, and characteristic polynomials. A related application of Cubic Equation Solver is rational candidates.
Boundaries and common traps
The leading coefficient cannot be zero. Numerical root finding may show tiny imaginary remnants near repeated or real roots, which are normalized by tolerance.
Reversing the displayed steps offers a quick independent check on this cubic equation solver result.
Follow one set of numbers
x³−6x²+11x−6 factors as (x−1)(x−2)(x−3), so its roots are 1,2,3.
Reproducing the answer
Normalize the polynomial, iteratively refine three complex root estimates, sort them, and substitute to assess residual size.
Using the result beyond this page
Start the cubic equation solver setup by pairing every source number with cubic coefficient a, quadratic coefficient b, linear coefficient c and constant d. Convert unlike units before typing, and postpone rounding until the displayed three roots is ready to report.
How the output responds
Small coefficient changes can separate a repeated root or move a conjugate pair off the real axis. Use that direction of change to check the displayed three roots before copying it elsewhere.
Precision and reporting
When copying the result elsewhere, include its label and any squared, linear, angular, or percentage unit implied by the inputs. That record distinguishes a calculated three roots from an unlabeled number and makes later checking substantially easier.
A rational-root search can expose exact factors; this solver returns all roots numerically. Keep that boundary in mind when interpreting the numerical result.
For later verification, record cubic coefficient a, quadratic coefficient b, linear coefficient c and constant d before rounding the three roots. The unrounded working value can feed subsequent steps while the rounded value serves presentation.
A second route through Cubic Equation Solver
Substitute every displayed root into the original coefficients and inspect the residual. Conjugate imaginary parts should appear with equal magnitude and opposite signs for real coefficients.
For Cubic Equation Solver, use this relationship with the page example first, then repeat it with the entered values. The result should retain the expected sign, units, and scale. A mismatch usually identifies a transposed field or a degree-versus-radian mistake before it suggests numerical instability.