Question : If $x+\frac{1}{x}=-2$, then what is the value of $x^7+x^{-7}+x^2+x^{-2} ?(\mathrm{x}<0)$
Option 1: 4
Option 2: 2
Option 3: 1
Option 4: 0
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Correct Answer: 0
Solution :
$x+\frac{1}{x}=-2$
$⇒x^2+1=-2x$
$⇒x^2+2x+1=0$
$⇒(x+1)^2 = 0$
$\therefore x = -1$
By putting $x=-1$, we get,
$x^7+x^{-7}+x^2+x^{-2}=(-1)^7+(-1)^{-7}+(-1)^2+(-1)^{-2} = -2+2 = 0$
Hence, the correct answer is 0.
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