(C) Let $y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3}$.$y(x^2 + 2x + 3) = x^2 + 14x + 9$$x^2(y - 1) + 2x(y - 7) + (3y - 9) = 0$.Since $x$ is real,the discri

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(C) Let $y = \frac{x^2 + 14x + 9}{x^2 + 2x + 3}$.$y(x^2 + 2x + 3) = x^2 + 14x + 9$$x^2(y - 1) + 2x(y - 7) + (3y - 9) = 0$.Since $x$ is real,the discriminant $D \geq 0$:$D = [2(y - 7)]^2 - 4(y - 1)(3y - 9) \geq 0$$4(y^2 - 14y + 49) - 4(3y^2 - 12y + 9) \geq 0$$y^2 - 14y + 49 - 3y^2 + 12y - 9 \geq 0$$-2y^2 - 2y + 40 \geq 0$$y^2 + y - 20 \leq 0$$(y + 5)(y - 4) \leq 0$.Thus,$y$ lies between $-5$ and $4$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Mix Examples-Quadratic Equations and Inequations

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If $x$ is real,then the value of the expression $\frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ lies between

A
$5$ and $4$
B
$5$ and $-4$
C
$-5$ and $4$
D
None of these

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