(D) The given quadratic expression is $f(x) = x^2 + 2bx + c$.Completing the square,we get $f(x) = (x + b)^2 + (c - b^2)$.For $f(x)$ to be always posit

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(D) The given quadratic expression is $f(x) = x^2 + 2bx + c$.Completing the square,we get $f(x) = (x + b)^2 + (c - b^2)$.For $f(x)$ to be always positive for all real $x$,the minimum value of the expression must be greater than $0$.The minimum value occurs at $x = -b$,which is $f(-b) = c - b^2$.Thus,we require $c - b^2 > 0$,which implies $b^2 < c$.

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

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For what condition is the value of the quadratic expression $x^2 + 2bx + c$ always positive?

A
$b^2 - 4c > 0$
B
$b^2 - 4c < 0$
C
$c^2 < b$
D
$b^2 < c$

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

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The number of values of $k,$ for which the equation $x^{2}-3x+k=0$ has two distinct roots lying in the interval $(0,1),$ is

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If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

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Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

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The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

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For what condition is the value of the quadratic expression $x^2 + 2bx + c$ always positive?

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The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

The number of values of $k,$ for which the equation $x^{2}-3x+k=0$ has two distinct roots lying in the interval $(0,1),$ is

If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

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