Question 996546
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Solve: sqrt x-3 = sqrt 2 - sqrt x for x 
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{{{sqrt(x-3)}}} = {{{sqrt(2)}}} - {{{sqrt(x)}}}.


Square both sides. You will get


{{{x - 3}}} = {{{2}}} - {{{2*sqrt(2)*sqrt(x)}}} + {{{x}}}.


Simplify:


{{{-3}}} - {{{2}}} =  - {{{2*sqrt(2)*sqrt(x)}}}


{{{5}}} = {{{2*sqrt(2)*sqrt(x)}}}.


Square both sides again:


25 = 8x.


x = {{{25/8}}}.


<U>Check</U>: &nbsp;Left side is &nbsp;&nbsp;{{{sqrt(x-3)}}} = {{{sqrt(25/8-3)}}} = {{{sqrt((25-24)/8)}}} = {{{sqrt(1/8)}}} = {{{1/(2*sqrt(2))}}}. 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Right side is &nbsp;&nbsp;{{{sqrt(2)}}} - {{{sqrt(x)}}} = {{{sqrt(2)}}} - {{{sqrt(25/8)}}} = {{{sqrt(2)}}} - {{{(5/(2*sqrt(2)))}}} = {{{(4-5)/(2*sqrt(2))}}} = {{{-1/(2*sqrt(2))}}}.


The results for the left side and the right side are different.


It means that the original equation has no solutions. &nbsp;(The solution that we obtained is the solution to the equation 

with the squared left and right sides of the original equation, &nbsp;but it doesn't satisfy the original equation. 

Again, &nbsp;it means that the original equation has no solutions).


The plots in the Figure below confirm this conclusion.


<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -5.5, 5.5, -5.5, 5.5,
          sqrt(x-3),
          sqrt(2)-sqrt(x),
          sqrt(x-3) - (sqrt(2)-sqrt(x))
)}}}


Plots: {{{sqrt(x-3)}}} &nbsp;(red); &nbsp;{{{sqrt(2)-sqrt(x)}}} &nbsp;(green); &nbsp;{{{sqrt(x-3) - (sqrt(2)-sqrt(x))}}} &nbsp;(blue).

  </TD>
  </TR>
</TABLE>