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  4. The number of real solutions of the equation 3(x2+(1/x2))-2(x+(1/x))+5=0, is

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Q. 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

JEE MainJEE Main 2023Complex Numbers and Quadratic Equations

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 $ No solution.