QuickCalculators solves a cubic equation ax³ + bx² + cx + d = 0 for its roots and reports how many real roots the discriminant pattern implies. Enter the four coefficients, then read the roots with a Vieta check on sums and products when verification is shown.
A cubic always has three roots counting multiplicity in the complex numbers. This page emphasizes the real roots students usually need and notes complex partners when they appear.
Solve a cubic equation
Solving a cubic means finding all x values that make ax³ + bx² + cx + d equal zero, with a ≠ 0. QuickCalculators uses a stable cubic solver path and presents roots with multiplicities when repeated roots occur. Depressed-cubic reductions and Cardano-style steps may appear in the work panel for teaching builds; the numeric roots remain the primary output.
Always keep a nonzero.
Find the roots of a cubic
A cubic has either three real roots (counting multiplicity) or one real root and two complex conjugate roots when coefficients are real. QuickCalculators lists each root and labels real versus complex cases clearly.
Example fixture x³ − 6x² + 11x − 6 = 0 factors as (x − 1)(x − 2)(x − 3) = 0, so the roots are 1, 2, and 3.
Multiple roots share the same value with multiplicity greater than one.
Understand the discriminant of a cubic
The cubic discriminant distinguishes three distinct real roots, a multiple real root, or one real root with a complex conjugate pair. QuickCalculators can summarize that case without requiring the student to expand the full discriminant polynomial by hand. A positive discriminant for the usual real-coefficient cubic corresponds to three distinct real roots.
A zero discriminant signals at least one repeated root. A negative discriminant signals one real root and two complex conjugates. Exact threshold wording follows the engine's discriminant definition on the page.
Check the roots with Vieta's formulas
Vieta's formulas relate elementary symmetric sums of the roots to the coefficients. For ax³ + bx² + cx + d = 0, the sum of roots equals −b/a, and the product of roots equals −d/a (with sign conventions matching the monic form). QuickCalculators can show those checks after listing roots.
For roots 1, 2, and 3 of x³ − 6x² + 11x − 6, the sum is 6 = −(−6)/1 and the product is 6 = −(−6)/1. Failed Vieta checks mean a root listing error.
Solve x³ − 6x² + 11x − 6 = 0
The fixture solves x³ − 6x² + 11x − 6 = 0 on QuickCalculators.
- Enter a = 1, b = −6, c = 11, d = −6.
- Obtain roots 1, 2, and 3.
- Verify: (1)(2)(3) = 6 and 1 + 2 + 3 = 6 match Vieta for this monic cubic.
- Expand (x − 1)(x − 2)(x − 3) to recover the original polynomial.
Avoid this common misconception
A common belief is that every cubic has three distinct real roots. With real coefficients, a cubic can have only one real root while the other two are complex conjugates. Graphically that means one real x-intercept. QuickCalculators states the real-versus-complex split so three real listings are not assumed.
Frequently asked questions
What is a cubic equation?
A cubic equation is a polynomial equation of degree 3, typically written ax³ + bx² + cx + d = 0 with a ≠ 0.
What are the roots of x³ − 6x² + 11x − 6 = 0?
The roots are 1, 2, and 3. Entering those coefficients returns that fixture.
How many roots does a cubic have?
A cubic has three roots counting complex roots and multiplicity. The number of distinct real roots may be one or three (counting a triple or double as repeated reals).
What is Vieta's formula for a cubic?
For ax³ + bx² + cx + d = 0, the sum of the roots is −b/a and the product of the roots is −d/a under the standard signed conventions for that form.
Can a cubic have repeated roots?
Yes. A repeated root has multiplicity greater than one and makes the discriminant zero. The graph is tangent to the x-axis at that root.
How do you check cubic roots?
Substitute each root into the polynomial, or use Vieta sum and product checks against the coefficients.
Summary
QuickCalculators solves ax³ + bx² + cx + d = 0 and reports real and complex roots with Vieta checks. The fixture x³ − 6x² + 11x − 6 = 0 has roots 1, 2, and 3. Not every real cubic shows three distinct real roots; one real root with a conjugate pair is common.
Keep a ≠ 0 and verify with substitution or Vieta.