Question 370486
<pre>
{{{sqrt(4-x) - sqrt(x+6) =2}}}

Isolate one of the radical terms:

{{{sqrt(4-x) =2+ sqrt(x+6)}}}

Square both sides:

{{{(sqrt(4-x))^2 =(2+ sqrt(x+6))^2}}}

Take away the radical and square on the left.
Write expression squared on the right as multiplied by itself:

{{{4-x =(2+ sqrt(x+6))(2+sqrt(x+6))}}}

Use FOIL on the right:

    F  O    I    L 
{{{4-x =4+ 2sqrt(x+6)+2sqrt(x+6)+ (sqrt(x+6))^2)}}}

Simplify:

{{{4-x =4+ 4sqrt(x+6)+ (x+6)}}}

{{{4-x =4+ 4sqrt(x+6)+ x+6}}}

{{{4-x = 10+ 4sqrt(x+6)+ x}}}

Isolate the radical term:

{{{-6-2x = 4sqrt(x+6)}}}

Divide all three terms by -2

{{{3+x = -2sqrt(x+6)}}}

Square both sides:

{{{(3+x)^2 = (-2sqrt(x+6))^2}}}

Write expression squared on the left as multiplied by itself 
Take away the radical and square on the right. 
Don't forget to square the -2 on the right as +4

{{{(3+x)(3+x) = 4(x+6)}}}

Use FOIL on the left, distribute on the right side:

{{{9+3x+3x+x^2 = 4x+24}}}

Simplify

{{{9+6x+x^2 = 4x+24}}}

Get 0 on the right

{{{x^2+2x-15= 0}}}

Factor

{{{(x+5)(x-3)= 0}}}

Use zero-factor principle:

{{{x+5=0}}} gives {{{x=-5}}}

{{{x-3=0}}} gives {{{x=3}}}

Test {{{x=-5}}} in the original equation to see if it 
is a correct solution or an extraneous one:

{{{sqrt(4-x) - sqrt(x+6) =2}}}

{{{sqrt(4-(-5)) - sqrt((-5)+6) =2}}}

{{{sqrt(4+5) - sqrt(-5+6) =2}}}

{{{sqrt(9) - sqrt(1) =2}}}

{{{3-1=2}}}

{{{2=2}}}

That checks, so {{{x=-5}}} is a correct solution.

Test {{{x=3}}} in the original equation to see if it 
is a correct solution or an extraneous one:

{{{sqrt(4-x) - sqrt(x+6) =2}}}

{{{sqrt(4-(3)) - sqrt((3)+6) =2}}}

{{{sqrt(4-3) - sqrt(3+6) =2}}}

{{{sqrt(1) - sqrt(9) =2}}}

{{{1-3=2}}}

{{{-2=2}}}

That does not check, so {{{x=3}}} is an extraneous solution and
must be discarded.

The only solution is {{{x=-5}}}

Edwin</pre>