State all the values of \(x\) that satisfy the following equation: \(\left(7-\frac{2}{x}\right)=\sqrt{\frac{7}{x}}\).
So far I can only find 1 solution to this, and would like to know if there are more.
Thanks!
+6251 \(\left(7-\dfrac 2 x\right) = \sqrt{\dfrac 7 x}\\ \left(7-\dfrac 2 x\right)^2 = \dfrac{7}{|x |} \\ 49-\dfrac{28}{x} + \dfrac{4}{x^2} = \dfrac{7}{|x|}\)
\(\text{let $x>0$}\\ 49-\dfrac{28}{x} + \dfrac{4}{x^2} = \dfrac{7}{x}\\ 49x^2 - 28x + 4 = 7x\\ 49x^2 - 35x+ 4 = 0\\ x = \dfrac{35\pm \sqrt{(35)^2 - (4)(49)(4)}}{98} = \left(\dfrac 4 7,~\dfrac 1 7\right)\\ \text{We do however notice that $x = \dfrac 1 7$ is not a solution of the original equation $\\$ so it is discarded leaving only $x = \dfrac 4 7$}\)
\(\text{let $x\leq0$}\\ 49-\dfrac{28}{x} + \dfrac{4}{x^2} = -\dfrac 7 x\\ 49x^2-21x+4 = 0\\~\\ \text{We note that this equation has no real solutions and thus would not satisfy the original equation}\)
\(\text{So the only solution is $x= \dfrac 4 7$}\)
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