Problem Statement:-
Solve the equation $$x^2+\dfrac{9x^2}{(x+3)^2}=27$$
I have tried to turn it into a quadratic equation so as to be saved from solving a quartic equation, but have not been able to come up with anything of much value.
These are the things that I have tried to turn the given equation into a quadratic equation.
$$x^2+\dfrac{9x^2}{(x+3)^2}=27\implies 1+\dfrac{1}{\left(\dfrac{x}{3}+1\right)^2}=3\left(\dfrac{3}{x}\right)^2$$ $$\text{OR}$$
$$x^2+\dfrac{9x^2}{(x+3)^2}=27\implies 1+\dfrac{\left(\dfrac{3}{x}\right)^2}{\left(1+\dfrac{3}{x}\right)^2}=3\left(\dfrac{3}{x}\right)^2$$
Best Answer
HINT:
Using $a^2+b^2=(a-b)^2+2ab,$ $$x^2+\left(\dfrac{3x}{x+3}\right)^2=\left(x-\dfrac{3x}{x+3}\right)^2+2\cdot x\cdot\dfrac{3x}{x+3}=\left(\dfrac{x^2}{x+3}\right)^2+6\cdot\dfrac{x^2}{x+3}$$
Generalization :
For $a^2+b^2=k$
If $\dfrac{ab}{a+b}=c$ where $c$ is a non-zero finite constant,
$$\implies k=(a+b)^2-2ab=(a+b)^2-2(a+b)c$$ $$\iff(a+b)^2-2(a+b)c-k=0$$ which isa Quadratic equation in $a+b$
Can you recognize $a,b$ here?
If $\dfrac{ab}{a-b}=c$ use $a^2+b^2=(a-b)^2+2ab$