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Two positive numbers differ by 11, and their square roots 
differ by 1.  Find the numbers.

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>>...Two positive numbers differ by 11,,,<<

  x - y = 11

>>...their square roots differ by 1...<<
  _    _
 Öx - Öy = 1

So we have to solve the system:

  x - y = 11
  _    _
 Öx - Öy = 1

We use substitutions. Solve the first
equation for x

  x - y = 11

      x = 11 + y

Then we substitute 11 + y for x in the second equation:

  ______    _
 Ö11 + y - Öy = 1

Isolate the most complicated radical term:
       ______        _
      Ö11 + y = 1 + Öy

Square both sides:

     ______           _
   (Ö11 + y)² = (1 + Öy)

To square the left we only need to take away the radical
because the left consists of only one radical term.

However to square the right, we must put it down two
times and multiply it out using FOIL
                      _       _ 
       11 + y = (1 + Öy)(1 + Öy)
                     _    _     _ 
       11 + y = 1 + Öy + Öy + (Öy)²
                      _
       11 + y = 1 + 2Öy + y

Isolate the radical term:
                  _
           10 = 2Öy

Divide both sides by 2
                 _ 
            5 = Öy

Square both sides
                  _
           5² = (Vy)²

           25 = y

To find the other number x, we substitute 25 for y in

            x = 11 + y
    
            x = 11 + 25

            x = 36

So the two numbers are 36 and 25.

Checking:

They differ by 36 - 25 or 11
Their square roots are 6 and 5
Their square roots differ by 6 - 5 or 1

Edwin
AnlytcPhil@aol.com