Cubic Equation Calculator

About Cubic Equation Calculator

Cubic equations can have one or three real roots. Solutions involve Cardano's formula and trigonometric methods.

How to use this calculator

1. Enter the coefficients \(a\), \(b\), \(c\), and \(d\) from your cubic equation.
2. Make sure the equation is written in the form \(ax^3 + bx^2 + cx + d = 0\).
3. Click calculate to find one real root.
4. Use the displayed steps to check how the root was found or to continue solving the equation.

The formula explained

The formula \(ax^3 + bx^2 + cx + d = 0\) describes a cubic equation. The calculator finds a value of \(x\) that makes the left side equal to 0, so that value is a real root.

• a = coefficient of \(x^3\), and it cannot be 0
• b = coefficient of \(x^2\)
• c = coefficient of \(x\)
• d = constant term
• x = the variable, whose root we want to find

Step by step method

1. Rewrite the equation in standard form, with all terms on one side: \(ax^3 + bx^2 + cx + d = 0\).
2. Test possible rational roots if they are easy to try, such as factors of the constant term divided by factors of the leading coefficient.
3. Once you find a value that makes the expression equal to 0, that value is one real root.

Worked example

Problem. Find one real root of \(x^3 - 6x^2 + 11x - 6 = 0\).

1. Try \(x = 1\): \(1^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0\).
2. Since the result is 0, \(x = 1\) is a root of the cubic.
3. So one real root is \(x = 1\).

Tips and common mistakes

• If your cubic does not start with \(x^3\), do not forget the leading coefficient \(a\). For example, \(2x^3 - 3x^2 + x - 5 = 0\) has \(a = 2\).
• A real root is only one solution. A cubic can have more roots, including repeated roots or nonreal roots.

Frequently asked questions