(B) Given $f(x) = ax^2 + 2bx + c$. Since $f(x) = 0$ has imaginary roots,the discriminant $D = (2b)^2 - 4ac < 0$,which implies $4b^2 - 4ac < 0$,or $b^2

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(B) Given $f(x) = ax^2 + 2bx + c$. Since $f(x) = 0$ has imaginary roots,the discriminant $D = (2b)^2 - 4ac Since $a \neq 0$,if $a > 0$,then $ac > b^2 \geq 0$,so $c > 0$. However,the graph of $f(x)$ would be an upward parabola,and since $D Therefore,we must have $a Since $a Thus,$f(x) Specifically,at $x = 0$,$f(0) = c$. Since the entire graph is below the $x$-axis,$f(0) < 0$,which means $c < 0$.

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

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?

A
$c > 0$
B
$c < 0$
C
$c = 0$
D
Data insufficient

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Class 11

Mathematics Questions

Chapter

4-2.Quadratic Equations and Inequations

Subtopic

Quadratic expressions and Position of roots

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 $x$?

Medium

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For what minimum value of $k$ do both roots of the equation $x^2 - 8kx + 16(k^2 - k + 1) = 0$ exist,are real,distinct,and at least $4$?

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Match the following: Consider the equation $x^2 + 2(a - 1)x + a + 5 = 0$. Match the real values of $a$ with the conditions on the roots of the given equation.

Column-$I$ Column-$II$

$A$. Imaginary roots $P$. $a \in (-1, 4)$

$B$. One root less than $3$ and other greater than $3$ $Q$. $a \in (-\infty, -1)$

$C$. One root less than $1$ and other greater than $3$ $R$. $a \in (-\infty, -4/3)$

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

Difficult

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Match the following quadratic expressions with their minimum values:
Quadratic expression The minimum value
i) $x^2 + 4x + 6$ a) $1$
ii) $x^2 - 2x + 5$ b) $2$
iii) $x^2 + 6x + 18$ c) $4$
iv) $x^2 - 4x + 5$ d) $9$

Quadratic expression	The minimum value
i) $x^2 + 4x + 6$	a) $1$
ii) $x^2 - 2x + 5$	b) $2$
iii) $x^2 + 6x + 18$	c) $4$
iv) $x^2 - 4x + 5$	d) $9$

DifficultAP EAMCET 2017

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

Easy

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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?

Similar Questions

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 $x$?

For what minimum value of $k$ do both roots of the equation $x^2 - 8kx + 16(k^2 - k + 1) = 0$ exist,are real,distinct,and at least $4$?

Match the following quadratic expressions with their minimum values:Quadratic expressionThe minimum valuei) $x^2 + 4x + 6$a) $1$ii) $x^2 - 2x + 5$b) $2$iii) $x^2 + 6x + 18$c) $4$iv) $x^2 - 4x + 5$d) $9$

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

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Match the following quadratic expressions with their minimum values:
Quadratic expression The minimum value
i) $x^2 + 4x + 6$ a) $1$
ii) $x^2 - 2x + 5$ b) $2$
iii) $x^2 + 6x + 18$ c) $4$
iv) $x^2 - 4x + 5$ d) $9$