(D) Let $f(x) = ax^2 + bx + c = a(x - \alpha)(x - \beta)$.Given that $\alpha < k < \beta$,the factors $(k - \alpha)$ and $(k - \beta)$ have opposite s

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(D) Let $f(x) = ax^2 + bx + c = a(x - \alpha)(x - \beta)$.Given that $\alpha Specifically,$(k - \alpha) > 0$ and $(k - \beta) For the expression $f(k) = a(k - \alpha)(k - \beta)$ to be negative,we must have $a(k - \alpha)(k - \beta) Multiplying by $a$,we get $a^2(k - \alpha)(k - \beta) < 0$,which is equivalent to $a(ak^2 + bk + c) < 0$ or $a^2k^2 + abk + ac < 0$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Quadratic expressions and Position of roots

Easy

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$ and $k$ is a real number,then the condition for $\alpha < k < \beta$ is given by:

A
$ac > 0$
B
$ak^2 + bk + c = 0$
C
$ac < 0$
D
$a(ak^2 + bk + c) < 0$

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

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If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$ and $k$ is a real number,then the condition for $\alpha < k < \beta$ is given by:

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