Question 707774
<pre>
    &#8730;<span style="text-decoration: overline">3x+4</span> - &#8730;<span style="text-decoration: overline">2x+4</span> = 2

Isolate either one of the radicals.
I'll isolate the first radical on the left side:

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

Square both sides:

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

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

FOIL out the right side

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

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

             3x+4 = 8 + 4&#8730;<span style="text-decoration: overline">2x+4</span> + 2x

Isolate the remaining radical term on the right:

              x-4 = 4&#8730;<span style="text-decoration: overline">2x+4</span>

Square both sides:

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

       (x-4)(x-4) = 16(2x+4)

         x²-8x+16 = 32x+64

        x²-40x-48 = 0

That doesn't factor so we have to use the quadratic formula:

                x = {{{(-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 

                x = {{{(-(-40) +- sqrt( (-40)^2-4*(1)*(-48) ))/(2*(1)) }}} 

                x = {{{(40 +- sqrt(1600+192 ))/2 }}}  

                x = {{{(40 +- sqrt(1792 ))/2 }}}
     
                x = {{{(40 +- sqrt(256*7 ))/2 }}}

                x = {{{(40 +- 16sqrt(7))/2 }}}

                x = {{{(8(5 +- 2sqrt(7)))/2 }}}

                x = 4(5 ± 2&#8730;<span style="text-decoration: overline">7</span>)

Using the +,    x = 4(5 + 2&#8730;<span style="text-decoration: overline">7</span>), approximately 41.16601049

Using the -,    x = 4(5 - 2&#8730;<span style="text-decoration: overline">7</span>), approximately -1.166010489

We must check for extraneous solutions:

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

Checking 41.2  (rounded to tenths)

&#8730;<span style="text-decoration: overline">3(41.2)+4</span> - &#8730;<span style="text-decoration: overline">2(41.2)+4</span> = 2
&#8730;<span style="text-decoration: overline">127.6</span> - &#8730;<span style="text-decoration: overline">86.4</span> = 2
11.296 - 9.295 = 2
         2.001 = 2

Not a perfect check, but close enough to believe it's 
a solution, sibce we rounded off.

x = 4(5 + 2&#8730;<span style="text-decoration: overline">7</span>), 

Checking -1.17  (rounded to hundredths)

&#8730;<span style="text-decoration: overline">3(-1.17)+4</span> - &#8730;<span style="text-decoration: overline">2(-1.17)+4</span> = 2
&#8730;<span style="text-decoration: overline">2(41.2)+4</span> - &#8730;<span style="text-decoration: overline">.49</span> = 2
   .7 - 1.66 = 2
        -.96 = 2

That's not close at all, so we believe it
is extraneous, and not a solution.

So there is just one solution:  x = 4(5 + 2&#8730;<span style="text-decoration: overline">7</span>).

Edwin</pre>