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Question

If $$\left( {a{x^2} + bx + c} \right)y + a'{x^2} + b'x + c' = {0^{}}$$, find the condition that $$x$$ may be a rational function of $$y$$.


Solution

$$(ax^{2}+bx+c)y+a^{'}x^{2}+b^{'}x+c^{'}=0$$

For, y is a rational function,
$$y=\cfrac{-(a^{'}x^{2}+b^{'}x+c^{'})}{ax^{2}+bx+c}$$

$$ \Rightarrow$$ The denominator, $$ax^{2}+bx+c\ne 0$$

$$ \Rightarrow x\ne \cfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$

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