(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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Class 11

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Chapter

4-2.Quadratic Equations and Inequations

Subtopic

Quadratic expressions and Position of roots

For the quadratic equation $x^2 - (K + 1)x + (K^2 + K - 8) = 0$,if one root is greater than $2$ and the other root is less than $2$,then $K$ lies in which interval?

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The values of $a$ for which the equation $2x^2 - 2(2a + 1)x + a(a + 1) = 0$ has one root less than $a$ and the other root greater than $a$ are given by:

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If the roots of the equation $x^2 - 2ax + a^2 + a - 3 = 0$ are real and less than $3$,then

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If the roots of ${x^2} + x + a = 0$ exceed $a$,then

Easy

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If $x$ is real,then the maximum value of $5 + 4x - 4x^2$ is equal to:

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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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For the quadratic equation $x^2 - (K + 1)x + (K^2 + K - 8) = 0$,if one root is greater than $2$ and the other root is less than $2$,then $K$ lies in which interval?

The values of $a$ for which the equation $2x^2 - 2(2a + 1)x + a(a + 1) = 0$ has one root less than $a$ and the other root greater than $a$ are given by:

If the roots of the equation $x^2 - 2ax + a^2 + a - 3 = 0$ are real and less than $3$,then

If the roots of ${x^2} + x + a = 0$ exceed $a$,then

If $x$ is real,then the maximum value of $5 + 4x - 4x^2$ is equal to:

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