Food for thought

# How not to solve a cubic... Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Problem

A student was trying to solve the cubic $x^3-2x-1=0.$

Unfortunately, they thought they were dealing with the quadratic $x^2-2x-1=0$ and used the quadratic formula. They got the two solutions $x=1\pm\sqrt{2}$, neither of which is a root of the original cubic.

Could this wrong method ever work?

Can you find a cubic equation $ax^3+bx+c=0$ where (at least) one of the roots of the corresponding quadratic $ax^2+bx+c=0$ is also a root of the cubic?