Find the roots of

,

the composition of where

, where

Let’s begin with an example.

Let , where

Then, . So, if we can find the values of x such that the value of is a root of , we will find the roots of .

By the quadratic equation,

So when is 2 or 6, will be 0.

From the quadratic equation, when and when

So, the 4 roots of are 1, 2, 5 and 6.



For the general case, let where . Our goal is to find the roots of .

Then,

This may look a little disconcerting, but this is nothing more than a quadratic equation of the form , where . Therefore, by the quadratic formula,

For simplicity, call the right side of the above equality . Thus, .

But,

So, and

Once again we have a quadratic equation that can be easily solved.

Thus, .

Extending this pattern we obtain:

When m = n, can be solved to find the roots of .

Using the quadratic equation,

where

So, has roots (assuming the radical is never 0) and can be written algebraically using the following recursive relation.

where

If we apply this to the example above: