Suppose a-b-c - R- a 0 and 4a - 6b - 9c - 0 and ax2 - bx - c - 0 does not have real roots- then

The correct option is B 1 Given :ax^2+bx+c=0 We know that #160; ab gt;0⇒ ba gt;0 when a and b are positive. ab gt;0⇒ ba ...

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Standard X

Mathematics

Nature of Roots

Suppose a,b...

Question

Suppose $a, b, c \in R$ , $a \neq 0$ and $4 a - 6 b + 9 c < 0$ and $a x^{2} + b x + c = 0$ does not have real roots, then

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Solution

The correct option is B $1$
Given $: a x^{2} + b x + c = 0$
We know that
$a b > 0 \Rightarrow \frac{b}{a} > 0$ when $a$ and $b$ are positive.
$a b > 0 \Rightarrow \frac{b}{a} > 0$ when $a$ and $b$ are negative.
Sum of the roots $= - \frac{b}{a} < 0$ (negative)
Given: $4 a + 2 b + c < 0$ by substituting $x = 2$ in $a x^{2} + b x + c = 0$
$∴ x = 2$ is a root of $a x^{2} + b x + c = 0$
Thus, there exists only one root.

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Similar questions

Let a,b,c be real numbers with a ≠ 0. Suppose a and 4a + 3b + 2c have the same sign. Then
equation ax² + bx + c = 0 cannot have both the roots in the interval.

Q. Statement 1 : If

f (x) = a x^{2} + b x + c

, where

a > 0, c < 0

and

b \in R

, then roots of

f (x) = 0

must be real and distinct .
Statement 2 : If

f (x) = a x^{2} + b x + c,

where

a > 0, b \in R, b \neq 0

and the roots of

f (x) = 0

are real and distinct, then

c

is necessarily negative real number .

Let a,b,c be real numbers with a ≠ 0. Suppose a and 4a + 3b + 2c have the same sign. Then
equation ax² + bx + c = 0 cannot have both the roots in the interval.

The equation $a x^{2} + b x + c$ = 0 does not have real roots and c < 0. Which of these is true?

Q. For

a x^{2} + b x + c = 0, a \neq 0, b = 0

, roots are real if and only if

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Suppose a,b,c∈R, a≠0 and 4a−6b+9c<0 and ax2+bx+c=0 does not have real roots, then

The correct option is B 1Given:ax2+bx+c=0We know that ab>0⇒ba>0 when a and b are positive.ab>0⇒ba>0 when a and b are negative.Sum of the roots=−ba<0(negative)Given:4a+2b+c<0 by substituting x=2 in ax2+bx+c=0∴x=2 is a root of ax2+bx+c=0Thus, there exists only one root.

Suppose $a, b, c \in R$ , $a \neq 0$ and $4 a - 6 b + 9 c < 0$ and $a x^{2} + b x + c = 0$ does not have real roots, then