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If $a+b+c=0$, then the roots of the equation $4 a x^2+3 b x+2 c=0$ are
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Real
We have $4 a x^2+3 b x+2 c=0$ Let roots are $\alpha$ and $\beta$
Let $D=B^2-4 A C=9 b^2-4(4 a)(2 c)=9 b^2-32 a c$
Given that, $\quad(a+b+c)=0 \Rightarrow b=-(a+c)$
Putting this value, we get
$=9(a+c)^2-32 a c=9(a-c)^2+4 a c$
Hence roots are real
Let $D=B^2-4 A C=9 b^2-4(4 a)(2 c)=9 b^2-32 a c$
Given that, $\quad(a+b+c)=0 \Rightarrow b=-(a+c)$
Putting this value, we get
$=9(a+c)^2-32 a c=9(a-c)^2+4 a c$
Hence roots are real
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