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4-2.Quadratic Equations and Inequations
hard
The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$, is
A
$4$
B
$0$
C
$3$
D
$2$
(JEE MAIN-2023)
Solution
$3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$
$3\left[\left(x+\frac{1}{x}\right)^2-2\right]-2\left(x+\frac{1}{x}\right)+5=0$
Let $x +\frac{1}{ x }= t$
$3 t ^2-2 t -1=0$
$3 t ^2-3 t + t -1=0$
$3 t ( t -1)+1( t -1)=0$
$( t -1)(3 t +1)=0$
$t =1,-\frac{1}{3}$
$x +\frac{1}{ x }=1,-\frac{1}{3} \Rightarrow \text { No solution. }$
Standard 11
Mathematics
