Cubic Equation Calculator for Real and Complex Roots
Solve ax cubed plus bx squared plus cx plus d equals zero for all values of x
| Root | Value |
|---|
Cubic equation calculator results list all three roots
Cubic equation calculator results list all three roots whether they are real numbers, repeated values, or complex conjugate pairs. The table below shows an example for the equation x cubed minus 6x squared plus 11x minus 6 equals zero.
| Coefficients | Equation | Roots |
|---|---|---|
| a=1, b=-6, c=11, d=-6 | x³ – 6x² + 11x – 6 = 0 | x = 1, 2, 3 |
| a=1, b=0, c=0, d=-8 | x³ – 8 = 0 | x = 2 (plus 2 complex) |
| a=1, b=-3, c=3, d=-1 | x³ – 3x² + 3x – 1 = 0 | x = 1, 1, 1 |
Cubic equations always have three roots in total
Cubic equations always have three roots in total when counted with multiplicity, including any complex roots. A cubic equation written as ax cubed plus bx squared plus cx plus d equals zero always has either three real roots, which may repeat, or one real root paired with two complex conjugate roots.
Solving cubic equations with no constant term
Solving cubic equations with no constant term means the d value equals zero, which allows x to be factored out immediately. This gives x equals zero as one root, and the remaining quadratic equation can be solved using the quadratic formula or factoring methods.
Solving cubic equations using Vieta’s formulas
Solving cubic equations using Vieta’s formulas relates the roots p, q and r directly to the coefficients. The sum of the roots equals negative b divided by a, the sum of products of pairs of roots equals c divided by a, and the product of all three roots equals negative d divided by a.
Using the cubic formula for complex solutions
Using the cubic formula for complex solutions applies a general algebraic method similar to the quadratic formula but more involved. The cubic formula uses a discriminant to determine whether the equation has three real roots or one real root and two complex roots, then computes each root using radicals and trigonometric or cube root operations.
Real world cubic equations appear in many fields
Real world cubic equations appear in many fields including physics for projectile motion with drag, engineering for stress and volume calculations, economics for cost and profit functions, and computer graphics for curve modeling with cubic splines.
Verifying a root by substitution
Verifying a root by substitution involves plugging a candidate value back into the original equation ax cubed plus bx squared plus cx plus d. If the result equals zero, the value is confirmed as a valid root of the cubic equation.
Frequently asked questions about cubic equation calculations
A cubic equation is a polynomial equation where the highest exponent of the variable is 3, written in standard form as a x cubed plus b x squared plus c x plus d equals zero, where a is not zero. The b, c and d terms can be zero, but the a x cubed term must always be present.
Every cubic equation has exactly three roots when counted with multiplicity, including complex roots. A cubic equation always has either one real root and two complex conjugate roots, or three real roots which may include repeated values.
If the d term is zero, x can be factored out of the equation, which means x equals zero is always one of the solutions. The remaining factor becomes a quadratic equation that can be solved using the quadratic formula or factoring.
Vieta’s formulas relate the roots of a cubic equation to its coefficients. For roots p, q and r, the sum p plus q plus r equals negative b over a, the sum of pairwise products pq plus qr plus rp equals c over a, and the product pqr equals negative d over a.
Yes, a cubic equation can have complex roots, but they always occur in conjugate pairs when the coefficients are real numbers. This means if one complex root has the form a plus bi, another root will be a minus bi, and the third root will be real.
To check if a value is a root, substitute it for x in the equation a x cubed plus b x squared plus c x plus d, and if the result equals zero, that value is a root. This method is often used after factoring to verify a solution.