Question 30998
<pre><b>square root of 3x-5 minus square root x+7 equals 2
<font size = 3>                    ____    ___
                   <font face = "symbol">Ö</font>3x-5 - <font face = "symbol">Ö</font>x+7 = 2
                          ____            ___ 
Isolate the radical term <font face = "symbol">Ö</font>3x-5 by adding <font face = "symbol">Ö</font>x+7 to both sides. 
                           ____        ___
                          <font face = "symbol">Ö</font>3x-5 = 2 + <font face = "symbol">Ö</font>x+7 

Square both sides:

                          ____          ___
                       (<font face = "symbol">Ö</font>3x-5)² = (2 + <font face = "symbol">Ö</font>x+7)² 

Careful here.  Since there is only ONE term in the parenthese on the left to be
squared, and since it is a square root, we can just take away the radical and
get 3x-5. But the right side id not so easy because there are TWO terms in the
parentheses.  We must write the parenbtheses on the right with TWO terms twice
and multiply using FOIL

                                        ___       ___
                           3x-5 = (2 + <font face = "symbol">Ö</font>x+7)(2 + <font face = "symbol">Ö</font>x+7)

To use FOIL on the right we have

F = 2×2 = 4
      ___
O = 2<font face = "symbol">Ö</font>x+7
      ___
I = 2<font face = "symbol">Ö</font>x+7
      ___
L = (<font face = "symbol">Ö</font>x+7)² = x+7

So we have

                           3x - 5 = F + O + I + L
                                          ___     ___ 
                           3x - 5 = 4 + 2<font face = "symbol">Ö</font>x+7 + 2<font face = "symbol">Ö</font>x+7 + x + 7     
                                           ___
                           3x - 5 = 11 + 4<font face = "symbol">Ö</font>x+7 + x
                                    ___ 
We again isolate the radical term 4<font face = "symbol">Ö</font>x+7
                                       ___
                           2x - 16 = 4<font face = "symbol">Ö</font>x+7

Evewry term can be divided by 2
                                       ___
                             x - 8 = 2<font face = "symbol">Ö</font>x+7

Square both sides:
                                        _____
                          (x - 8)² = (2<font face = "symbol">Ö</font>x + 7)²
                                         _____
                    (x - 8)(x - 8) = 2²(<font face = "symbol">Ö</font>x + 7)²
             
                 x² - 8x - 8x + 64 = 4(x + 7)

                     x² - 16x + 64 = 4x + 28

                     x² - 20x + 36 = 0

                   (x - 18)(x - 2) = 0 

                   Answers: x = 18, x =2.
  
But we must ceck these in the original, because they both might be solutions,
one might be a solution and the other not, or there may be no solutions at all.
We substitute in the original equation:
                       ____    ___
                      <font face = "symbol">Ö</font>3x-5 - <font face = "symbol">Ö</font>x+7 = 2  

                   _______    ____
                  <font face = "symbol">Ö</font>3(18)-5 - <font face = "symbol">Ö</font>18+7 = 2
                          __    __
                         <font face = "symbol">Ö</font>49 - <font face = "symbol">Ö</font>25 = 2

                             7 - 5 = 2

                                 2 = 2    

This checks so x = 18 is a solution
                       ____    ___
                      <font face = "symbol">Ö</font>3x-5 - <font face = "symbol">Ö</font>x+7 = 2  

                     ______    ____
                    <font face = "symbol">Ö</font>3(2)-5 - <font face = "symbol">Ö</font>2+7 = 2
                            _    _
                           <font face = "symbol">Ö</font>1 - <font face = "symbol">Ö</font>9 = 2

                             1 - 3 = 2

                                -2 = 2

This does not check so x = 2 is NOT a solution                              

Edwin
AnlytcPhil@aol.com</pre>