If the equation $ax^{2} + bx + c = x,a,b,c \in R$ and $a
eq 0$, has no real roots, then the equati

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If the equation $ax^{2} + bx + c = x,a,b,c \in R$ and $a 
eq 0$, has no real roots, then the equation $a\left( ax^{2} + bx + c ight)^{2} + b\left( ax^{2} + bx + c ight) + c = x$ will have

Accepted Answer

Let $P(x) = {ax}^{2} + bx + c$$P(x) = x$ has no real roots$$\begin{matrix} & \ \Rightarrow \ P(x) > x\forall x \in R \ & \ \Rightarrow \ P(P(x)) > P(x) > x\forall x \in R \end{matrix}\ ext{~or~}\ \begin{matrix} & P(x) < x \ & P(P(x)) < P(x) < x\forall x \in R \end{matrix}\ \forall x \in R$$Hence, $P(P(x)) = x$ has no real roots

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