One positive number is 8 more than another.
The sum of their squares is 130.  Find both numbers.

Let one positive number = x
Let another positive number = y

>>...One positive number is 8 more than another...<<

So                    x = y + 8

>>...The sum of their squares is 130...<<

So                  x² + y² = 130      

We have the system of equations

x = y + 8
x² + y² = 130

Substitute (y + 8) from the first for x in the second

           (y + 8)² + y² = 130

     (y + 8)(y + 8) + y² = 130

  y² + 8y + 8y + 64 + y² = 130
 
            2y² + 16y + 64 = 130

            2y² + 16y - 66 = 0

           2(y² + 8y - 33) = 0

          2(y + 11)(y - 3) = 0   

Set each variable factor = 0

        y + 11 = 0 gives y = -11

         y - 3 = 0 gives y = 3

Discard the negative answer
 
Substitute 3 for y in 

             x = y + 8

             x = 3 + 8

             x = 11

So the numbers are 11 and 3

Edwin
AnlytcPhil@aol.com