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  4. The only real value of x satisfying the equation is 6 √(x/x+4)-2 √(x+4/x)=11(1) where x ∈ R

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Q. The only real value of $x$ satisfying the equation is $6 \sqrt{\frac{x}{x+4}}-2 \sqrt{\frac{x+4}{x}}=11$(1) where $x \in R$

Complex Numbers and Quadratic Equations

Solution:

For (1) to be defined either $x<-4$ or $x>0$.
Put $\sqrt{\frac{x}{x+4}}=y$. Equation (1) becomes
$ 6 y-2 / y=11 \Rightarrow 6 y^2-11 y-2=0 $
$\Rightarrow (6 y+1)(y-2)=0 \Rightarrow y=-1 / 6, y=2 $
$\text { As } y=\sqrt{\frac{x}{x+4}} \geq 0 \text {, we reject } y=-\frac{1}{6} . $
$\text { Thus, } y=2 \Rightarrow \frac{x}{x+4}=y^2=4$
$\Rightarrow x=4 x+16 \Rightarrow x=-16 / 3<-4$