Question 703976
<pre>
sole & check for extraneous solutions. 
X^1/2-(x-5)^1/2=2 
some how the answer is 81/16 but i dont know how to solve it
************************************************************<font face = tahoma><font size = 3><font color = blue><b>
The other person's solutions, 
"x≈5.055
or
x≈-1.055(reject, x≥0)" are WRONG. 
Although you provided the correct answer, he/she failed to match it.
Why bother to respond when you can't provide help, as needed? It's a SHAME!!!
                 {{{matrix(2,1, " ", x^(1/2) - (x - 5)^(1/2) = 2)}}}
                         {{{sqrt(x) - sqrt(x - 5) = 2}}} ----- Converting {{{1/2-power}}} to radical (square root), or {{{matrix(2,1, " ", a^(1/2))}}} to {{{sqrt(a)}}} 
The SMALLER radicand, x - 5, CANNOT be negative (< 0). So, {{{x - 5 >= 0}}} ==> {{{x >= 5}}}. We now have:
                        {{{sqrt(x) - sqrt(x - 5) = 2}}}, with {{{x >= 5}}}
{{{(sqrt(x))^2 - 2sqrt(x)sqrt(x - 5) + (sqrt(x - 5))^2 = 2^2}}} ---- Squaring each side
           {{{x - 2sqrt(x(x - 5)) + x - 5 = 4}}}
             {{{2x - 2sqrt(x^2 - 5x) - 5 = 4}}}
                    {{{- 2sqrt(x^2 - 5x) = 4 - 2x + 5}}}
                    {{{- 2sqrt(x^2 - 5x) = 9 - 2x}}}
               {{{(- 2sqrt(x^2 - 5x))^2 = (9 - 2x)^2}}} ---- Squaring each side
                      {{{4(x^2 - 5x) = 81 - 36x + 4x^2}}}
                        {{{4x^2 - 20x = 81 - 36x + 4x^2}}}
        {{{4x^2 - 4x^2 - 20x + 36x = 81}}}
                              16x = 81
                                  {{{highlight(matrix(1,3, x = 81/16, or, 5&1/16))}}} <=== This DEFINITELY matches your answer!

Per the above constraint, {{{x >= 5}}}, and {{{x = 81/16 = 5&1/16 >= 5}}}.
So, {{{81/16}}} is a VALID and ACCEPTABLE value fox x, in this instance!</pre>