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

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Solution

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

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

$\Rightarrow$ The denominator, $a x^{2} + b x + c \neq 0$

$\Rightarrow x \neq \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

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