Question 817886
<pre>
&#8730;<span style="text-decoration: overline">2x+3</span> - &#8730;<span style="text-decoration: overline">x+1</span> = 1

Isolate one of the radicals:

       &#8730;<span style="text-decoration: overline">2x+3</span> = 1-&#8730;<span style="text-decoration: overline">x+1</span>

Square both sides

    (&#8730;<span style="text-decoration: overline">2x+3</span>)² = (1-&#8730;<span style="text-decoration: overline">x+1</span>)²
 
Squaring the left side takes away the radical
Squaring the right side means to write the 
(expression twice
    
        2x+3 = (1-&#8730;<span style="text-decoration: overline">x+1</span>)(1-&#8730;<span style="text-decoration: overline">x+1</span>)

FOIL out the right side:

        2x+3 = 1-&#8730;<span style="text-decoration: overline">x+1</span>-&#8730;<span style="text-decoration: overline">x+1</span>+(&#8730;<span style="text-decoration: overline">x+1</span>)²

Simplify

        2x+3 = 1-2&#8730;<span style="text-decoration: overline">x+1</span>+x+1
    
        2x+3 = 2+x-2&#8730;<span style="text-decoration: overline">x+1</span>

Isolate the radical term

         x+1 = -2&#8730;<span style="text-decoration: overline">x+1</span>

Square both sides:

      (x+1)² = (-2&#8730;<span style="text-decoration: overline">x+1</span>)²

  (x+1)(x+1) = (-2)²(&#8730;<span style="text-decoration: overline">x+1</span>)²


    x²+x+x+1 = 4(x+1)

     x²+2x+1 = 4x+4

Get 0 on the right:

     x²-2x-3 = 0 

Factor the left side:

  (x-3)(x+1) = 0

Us zero-factor property:

  x-3 = 0;  x+1 = 0
    x = 3;    x = -1

We must check for extraneous solutions:

Checking x = 3      

    &#8730;<span style="text-decoration: overline">2x+3</span> - &#8730;<span style="text-decoration: overline">x+1</span> = 1
&#8730;<span style="text-decoration: overline">2(3)+3</span> - &#8730;<span style="text-decoration: overline">(3)+1</span> = 1
     &#8730;<span style="text-decoration: overline">6+3</span> - &#8730;<span style="text-decoration: overline">3+1</span> = 1
         &#8730;<span style="text-decoration: overline">9</span> - &#8730;<span style="text-decoration: overline">4</span> = 1
           3 - 2 = 1
               1 = 1

therefore x = 3 is a solution:

Checking x = -1      

      &#8730;<span style="text-decoration: overline">2x+3</span> - &#8730;<span style="text-decoration: overline">x+1</span> = 1
 &#8730;<span style="text-decoration: overline">2(-1)+3</span> - &#8730;<span style="text-decoration: overline">(1)+1</span> = 1
     &#8730;<span style="text-decoration: overline">-2+3</span> - &#8730;<span style="text-decoration: overline">-1+1</span> = 1
           &#8730;<span style="text-decoration: overline">1</span> - &#8730;<span style="text-decoration: overline">0</span> = 1
             1 - 0 = 1
                 1 = 1

Therefore x = -1 is also a solution.

Edwin</pre>