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# Values of x

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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!

Sep 10, 2019

#1
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$$\left(7-\dfrac 2 x\right) = \sqrt{\dfrac 7 x}\\ \left(7-\dfrac 2 x\right)^2 = \dfrac{7}{|x |} \\ 49-\dfrac{28}{x} + \dfrac{4}{x^2} = \dfrac{7}{|x|}$$

$$\text{let x>0}\\ 49-\dfrac{28}{x} + \dfrac{4}{x^2} = \dfrac{7}{x}\\ 49x^2 - 28x + 4 = 7x\\ 49x^2 - 35x+ 4 = 0\\ x = \dfrac{35\pm \sqrt{(35)^2 - (4)(49)(4)}}{98} = \left(\dfrac 4 7,~\dfrac 1 7\right)\\ \text{We do however notice that x = \dfrac 1 7 is not a solution of the original equation \\ so it is discarded leaving only x = \dfrac 4 7}$$

$$\text{let x\leq0}\\ 49-\dfrac{28}{x} + \dfrac{4}{x^2} = -\dfrac 7 x\\ 49x^2-21x+4 = 0\\~\\ \text{We note that this equation has no real solutions and thus would not satisfy the original equation}$$

$$\text{So the only solution is x= \dfrac 4 7}$$

.
Sep 10, 2019