The sum of two numbers is 12 and the sum of their 
squares is 78. Find the numbers. 

I do not believe that either number is whole.

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Let x = one number
Let y = other number

            x  + y  = 12
            x² + y² = 78

Solve the first equation for y

                  y = 12-x

Substitute (12-x) for y in the second equation:

            x² + y² = 78
       x² + (12-x)² = 78
x² + 144 - 24x + x² = 78

2x² - 24x + 66 = 0

x² - 12x + 33 = 0
         _
x = 6 ± Ö3
              _                    _       _
When x = 6 + Ö3, y = 12-x = 12-(6+Ö3) = 6-Ö3 

So one solution is 
             _     _
(x, y) = (6+Ö3, 6-Ö3)

              _                    _       _
When x = 6 - Ö3, y = 12-x = 12-(6-Ö3) = 6+Ö3 

So the other solution is 
             _     _
(x, y) = (6-Ö3, 6+Ö3)

Edwin