(C) For a quadratic expression $f(x) = ax^{2} + bx + c$ to have the same sign as $a$ for all $x$,the parabola must not intersect the $x$-axis.This imp

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(C) For a quadratic expression $f(x) = ax^{2} + bx + c$ to have the same sign as $a$ for all $x$,the parabola must not intersect the $x$-axis.This implies that the quadratic equation $ax^{2} + bx + c = 0$ has no real roots.For no real roots,the discriminant $D = b^{2} - 4ac$ must be less than $0$.Therefore,the condition is $b^{2} - 4ac < 0$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Solution of quadratic equations and Nature of roots

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The expression $ax^{2} + bx + c$ (where $a, b,$ and $c$ are real numbers) has the same sign as that of $a$ for all $x \in \mathbb{R}$ if:

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A
$b^{2} - 4ac > 0$
B
$b^{2} - 4ac \neq 0$
C
$b^{2} - 4ac < 0$
D
$b$ and $c$ have the same sign as that of $a$

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4-2.Quadratic Equations and Inequations

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Solution of quadratic equations and Nature of roots

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The expression $ax^{2} + bx + c$ (where $a, b,$ and $c$ are real numbers) has the same sign as that of $a$ for all $x \in \mathbb{R}$ if:

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