Question 46750
<pre><font size = 5><b>
 ___    ____
<font face = "symbol">Ö</font>x+3 + <font face = "symbol">Ö</font>2x-3 = 6

Isolate either radical term
        ____        ___ 
       <font face = "symbol">Ö</font>2x-3 = 6 - <font face = "symbol">Ö</font>x+3

Square both sides:
        ____           ___ 
      (<font face = "symbol">Ö</font>2x-3)² = (6 - <font face = "symbol">Ö</font>x+3)²

It's easy to square the left side, for the squaring
just cancels the square root.  However it's not so
easy to square the right side.  Put it down twice
and use FOIL
                       ___       ___
          2x-3 = (6 - <font face = "symbol">Ö</font>x+3)(6 - <font face = "symbol">Ö</font>x+3)

On the right:

"F" = 6·6 = 36
            ___       ___
"O" = (6)(-<font face = "symbol">Ö</font>x+3) = -6<font face = "symbol">Ö</font>x+3
         ___          ___
"I" = (-<font face = "symbol">Ö</font>x+3)(6) = -6<font face = "symbol">Ö</font>x+3
         ___    ___       ___
"L" = (-<font face = "symbol">Ö</font>x+3)(-<font face = "symbol">Ö</font>x+3) = (-<font face = "symbol">Ö</font>x+3)² = x+3 

So we have:
                       ___     ___
         2x-3 = 36 - 6<font face = "symbol">Ö</font>x+3 - 6<font face = "symbol">Ö</font>x+3 + x+3
                        ___
         2x-3 = 36 - 12<font face = "symbol">Ö</font>x+3 + x+3
                        ___
         2x-3 = 39 - 12<font face = "symbol">Ö</font>x+3 + x

Isolate the radical term
          ___
       12<font face = "symbol">Ö</font>x+3 = 39 + x - 2x + 3
          ___
       12<font face = "symbol">Ö</font>x+3 = 42 - x 

Square both sides:
         ___
     (12<font face = "symbol">Ö</font>x+3)² = (42 - x)²

      144(x+3) = (42 - x)(42 - x)

    144x + 432 = 1764 - 42x - 42x + x²

    144x + 432 = 1764 - 84x + x²

Get 0 on the left:

             0 = x² - 228x + 1332

The right side may factor. However since the
numbers are so big, it's probably easier to use
the quadratic formula:
                __________________
     -(-228) ± <font face = "symbol">Ö</font>(-228)²-4(1)(1332)
x = -------------------------------
                2(1)
             _____
      228 ± <font face = "symbol">Ö</font>46656
x = ----------------
            2
             
      228 ± 216
x = -------------
          2
              
Using the +

      228 + 216
x = -------------
          2

     444
x = ----- = 222
      2

Using the -

      228 - 216
x = -------------
          2

     12
x = ---- = 6
      2

Now we must check, because often we get extraneous 
answers in equations with even roots. Checking the
answer x = 222

           ___    ____
          <font face = "symbol">Ö</font>x+3 + <font face = "symbol">Ö</font>2x-3 = 6
     _____    ________
    <font face = "symbol">Ö</font>222+3 + <font face = "symbol">Ö</font>2(222)-3 = 6
            ___    ___
           <font face = "symbol">Ö</font>225 + <font face = "symbol">Ö</font>441 = 6

               15 + 21 = 6
 
                    36 = 6

No that doesn't check. So we discard that answer.
Checking the answer x=6

           ___    ____
          <font face = "symbol">Ö</font>x+3 + <font face = "symbol">Ö</font>2x-3 = 6
         ___    ______
        <font face = "symbol">Ö</font>6+3 + <font face = "symbol">Ö</font>2(6)-3 = 6
                _    _
               <font face = "symbol">Ö</font>9 + <font face = "symbol">Ö</font>9 = 6

                 3 + 3 = 6
 
                     6 = 6

That checks.  So there is one solution,
 
x = 6

Edwin</pre>