Cubic Equation

Tags

math

Is an equation of the form \( y = ax^3 + bx^2 + cx + d \).

Solving a Cubic

The common approach to finding the roots of a cubic is to find one root, then factorise it out of the expression and solve the remaining quadratic. How we find the root can be done through an exhaustive search or perhaps plotting it and observing where the curve falls to 0.

For example try solving \( x^3 + 3x^2 + 7x + 5 = 0 \), given that we know \( f(-1) = 0 \). From this we know \( x = -1 \) is a root so \( (x+1) \) must be a factor of this quadratic.

If we factorise it (for example using partial fractions) we get \( (x+1)(1 x^2 + 2x + 5) \). The inner term can be refactored into \( (x+1)^2 + 4 = 0 \) meaning the remaining roots of this cubic are \( x = -1 \pm 2i \).