Question

If $a,b,c$ are real distinct numbers satisfying the condition $a+b+c=0$, then the roots of the quadratic equation $3a{x}^2+5bx+7c=0$ are

• A. positive
• B. negative
• C. real and distinct
• D. imaginary

Answer 1

The correct option is C real and distinct

Given equation is : $3a{x}^2+5bx+7c=0$
Also given is that $a+b+c=0\dots \left(1\right)$
Discriminant of the equation $\Rightarrow D=b^2-4ac$

$={\left(5b\right)}^2-4\left(3a\right)\left(7c\right)$

$=25b^2-84ac\dots \left(2\right)$

Substituting the value of b from equation (1) in equation (2),we get

D $=25{(-a-c)}^2-84ac$

$=25{(a+c)}^2-84ac$

$=25a^2+25c^2-34ac$

$=4a^2+4c^2+8ac$ + $21a^2+{21c}^2-42ac$

$=4{(a+c)}^2+21{(a-c)}^2>0$
Since D>0,both the roots will be real and distinct.

Also,in the absence of information on the coefficients a,b and c in the equation we can't conclude whether both the roots will be positive or negative.

Hence,option (C) is the correct alternative.