One of the simplest problems is that of a quadratic with a large root and a small root, such as

x^{2} - 1E9*x + 1 = 0

The roots are 1E9 and 1E-9 and you will very likely get 1E9 and zero from most codings of a quadratic root solver such as the old formula of (1/2a)(-b±sqrt(b**2-4ac)).

Anyone should be able to get the roots of

x^{2} - 4*x + 4 = 0

quite easily as they are both equal to 2.

But, if we make a very small change to this equation

x^{2} - 4*x + 3.999999 = 0

the roots become 2.001 and 1.999. This is an example where a small relative change in the coefficients induces a much larger change in the roots.

A similar problem involving cubics is

x^{3} - 1E9*x^{2} + 3E9*x -2E9 = 0

Here, the roots are 1,2 and 1E9. Your old solver may compute that the smaller roots are zero, or that they are not accurately spaced.