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  4. The expression ( x 2+2 x +1/ x 2+2 x +7) lies in the interval ; ( x ∈ R ) -

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Q. The expression $\frac{ x ^2+2 x +1}{ x ^2+2 x +7}$ lies in the interval ; $( x \in R )$ -

Complex Numbers and Quadratic Equations

Solution:

$\frac{x^2+2 x+1}{x^2+2 x+7}=y$
$\Rightarrow x^2(1-y)+x(2-2 y)+1-7 y=0$
$D>0$
$\Rightarrow 6 y(y-1)<0 \Rightarrow y \in[0,1)$