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4-2.Quadratic Equations and Inequations
hard
The sum of all the real roots of the equation $\left( e ^{2 x }-4\right)\left(6 e ^{2 x }-5 e ^{ x }+1\right)=0$ is
A
$\log _{ c } 3$
B
$-\log _{e} 3$
C
$\log _{ e } 6$
D
$-\log _{e} 6$
(JEE MAIN-2022)
Solution
$\left( e ^{2 x}-4\right)\left(6 e ^{2 x }-3 e ^{ x }-2 e ^{ x }+1\right)=0$
$\left( e ^{2 x }-4\right)\left(3 e ^{ x }-1\right)\left(2 e ^{ x }-1\right)=0$
$e ^{2 x }=4 \text { or }^{ x }=\frac{1}{3} \text { or }^{ x }=\frac{1}{2}$
$\Rightarrow \text { sum of real roots }=\frac{1}{2} \ln 4+\ln \frac{1}{3}+\ln \frac{1}{2}$
$=-\ln 3$
Standard 11
Mathematics
