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  4. The number of real roots of the equation e6 x-e4 x-2 e3 x-12 e2 x+ex+1=0 is:

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Q. The number of real roots of the equation $e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0$ is:

JEE MainJEE Main 2021Complex Numbers and Quadratic Equations

Solution:

$e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0$
$\Rightarrow\left(e^{3 x}-1\right)^{2}-e^{x}\left(e^{3 x}-1\right)=12 e^{2 x}$
$\left(e^{3 x}-1\right)^{2}\left(e^{x}-e^{-x}-e^{-2 x}\right)=12$
$\Rightarrow e^{x}-e^{-x}-e^{-2 x}=\frac{12}{e^{3 x}-1}$
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$\Rightarrow$ No. of real roots $=2$