Cubic Equation Calculator

Instructions:

• Enter coefficients a, b, c, and d for the cubic equation.
• Click "Calculate" to find the roots of the cubic equation.
• Results will display the roots along with detailed calculations and explanations.
• Your calculation history will be shown below.
• Click "Clear Results" to reset the calculator.
• Click "Copy Results" to copy the results to the clipboard.

x^3 +

x^2 +

x +

Calculate Clear Results Copy Results

Detailed Calculation and Explanation:

Calculation History

Cubic Equation Calculator is a tool that helps to solve cubic equations. A cubic equation is an algebraic equation with a degree of 3. This means that the highest exponent in the equation is 3. Written in standard form, where a ≠ 0, a cubic equation looks like this: ax^3 + bx^2 + cx + d = 0. The b, c or d terms may be missing from the equation, or the a term might be 1. You have a cubic equation as long as there is an ax^3 value.

Concepts

The following are some of the key concepts that underlie cubic equations:

Roots

The solutions of a cubic equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:

• Algebraically: More precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.
• Trigonometrically: Numerical approximations of the roots can be found using root-finding algorithms such as Newton’s method.

Vieta’s Formulas

Vieta’s formulas show the relationship between the coefficients of a polynomial and the sums and products of its roots. If you know one root, you can do substitutions and figure out the others. For a cubic equation ax^3 + bx^2 + cx + d = 0, let p, q, and r be the 3 roots of the equation. So: (x − p)(x − q)(x − r) = 0, just as ax^3 + bx^2 + cx + d = 0. Vieta’s formulas use these equivalences to show how the roots relate to the coefficients of the cubic equation. The equivalences are listed below, along with the proof.

Vieta’s Equivalents	Root Expression	Equals
p + q + r	-b/a
pq + qr + rp	c/a
pqr	-d/a

Benefits

Cubic Equation Calculator is a useful tool for solving cubic equations. It can save time and effort compared to solving the equation by hand. The calculator can find all solutions for x, including complex solutions. There are either one or three possible real root solutions for x for any cubic equation. You may have only two distinct solutions as in the case x = 1, x = 5, x = 5, however, there are still three real roots.

Interesting Facts

• Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians.
• The problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed.
• In the 5th century BC, Hippocrates reduced this problem to finding two mean proportionals between one line and another twice its length, but could not solve this with a compass and straightedge construction. This task is now known to be impossible.

References

• Archimedes: On the Sphere and Cylinder, Book II, Proposition II
• Isaac Newton: Principia Mathematica, Book I, Proposition X
• Leonhard Euler: Introductio in Analysin Infinitorum, Volume I, Chapter 9
• Carl Friedrich Gauss: Disquisitiones Generales circa Superficies Curvas, Chapter 11