Hello there.
First, assume the numbers such that they satisties both affirmations:
- The sum of the squares of two numbers is
.
- The product of the two numbers is
.
With these informations, we can set the following equations:
Multiply both sides of the second equation by a factor of :
Make
We can rewrite the expression on the left hand side using the binomial expansion in reverse: , such that:
The square of a number is equal to if and only if such number is equal to
, thus:
Substituting that information from in
, we get:
Calculate the square root on both sides of the equation:
Once again with the information in , we have that:
The set of solutions of that satisfies both affirmations is:
This is the set we were looking for.