Question 637713

 What is the best way to solve this equation in step by step ?

{{{sqrt(2/x) - sqrt (x/2) = 1/sqrt (2) }}}

I'm profoundly grateful for help.


{{{sqrt(2/x) - sqrt (x/2) = 1/sqrt (2) }}}


{{{sqrt(2)/sqrt(x) - sqrt(x)/sqrt(2) = 1/sqrt (2) }}} ---- Separate the radicals 


{{{(sqrt(2)*sqrt(2)) - (sqrt(x)*sqrt(x)) = sqrt(x)}}} ----- Multiplying equation by LCD, {{{sqrt(x)*sqrt(2)}}}


{{{sqrt(2^2) - sqrt(x^2) = sqrt(x)}}} 


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


{{{(2 - x)^2 = (sqrt(x))^2}}} ----- Squaring both sides of equation


{{{4 - 4x + x^2 = x}}}


{{{x^2 - 4x - x + 4 = 0}}}


{{{x^2 - 5x + 4 = 0}}}


(x - 4)(x  - 1) = 0


x = 4 or x = 1


Substituting 4 for x in original equation results in x = 4 being an EXTRANEOUS solution. On the other hand, substituting 1 for x makes the original equation true, so {{{highlight_green(x=1)}}}.


You can do the checks by plugging in 4 for x, and then 1 for x in original equation.


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