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If $\alpha, \beta$ be the roots of the quadratic equation $a x^2+b x+c=0$ and $k$ be a real number, then the condition so that $\alpha \lt k \lt \beta$ is given by
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The correct answer is:
$a^2 k^2+a b k+a c \lt 0$
Here $a x^2+b x+c=a(x-\alpha)(x-\beta)$
Since $\alpha, \beta$ be the roots of $a x^2+b x+c=0$.
Also $\alpha \lt k \lt \beta$, so $a(k-\alpha)(k-\beta) \lt 0$
Also $a^2 k^2+a b k+a c=a\left(a k^2+b k+c\right)$
$=a^2(k-\alpha)(k-\beta) \lt 0 \Rightarrow a^2 k^2+a b k+a c \lt 0$
Since $\alpha, \beta$ be the roots of $a x^2+b x+c=0$.
Also $\alpha \lt k \lt \beta$, so $a(k-\alpha)(k-\beta) \lt 0$
Also $a^2 k^2+a b k+a c=a\left(a k^2+b k+c\right)$
$=a^2(k-\alpha)(k-\beta) \lt 0 \Rightarrow a^2 k^2+a b k+a c \lt 0$
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