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Q. Consider $f(x)=\left(x^2-3 x+2\right)^2-2\left(x^2-x-2\right)(x-1)^2+3\left(x^2-1\right)^2, x \in R$. Identify which of the following statement(s) is / are correct?

Complex Numbers and Quadratic Equations

Solution:

$ f ( x )=( x -1)^2( x -2)^2-2( x -2)( x +1)( x -1)^2+3( x -1)^2( x +1)^2$
$f (1)=0 \Rightarrow x =1$ is the solution of the equation $f ( x )=0$
$ \frac{f(x)}{\left(x^2-1\right)^2}=\left(\frac{x-2}{x+1}\right)^2-2\left(\frac{x-2}{x+1}\right)+3 $
$ g(t)=t^2-2 t+3, \text { where } t=\frac{x-2}{x+1}, t \neq 1,-\frac{1}{2} $
$ =(t-1)^2+2$
$\therefore g(t)>2$