Question : If $x$4 + $x$ -4 = 194, x > 0, then what is the value of $x+\frac{1}{x}+2$?
Option 1: 8
Option 2: 14
Option 3: 6
Option 4: 4
Correct Answer: 6
Solution : Given: $x^4+x^{-4}=194$ ⇒ $x^4+\frac{1}{x^4}=194$ Adding 2 both sides, we get: ⇒ $x^4+\frac{1}{x^4}+2=194+2$ ⇒ $(x^2+\frac{1}{x^2})^2=196$ ⇒ $x^2+\frac{1}{x^2}=14$ Adding 2 both sides, we get: ⇒ $x^2+\frac{1}{x^2}+2=14+2$ ⇒ $(x+\frac{1}{x})^2=16$ ⇒ $x+\frac{1}{x}=4$ Now, $x+\frac{1}{x}+2$ = 4 + 2 = 6 Hence, the correct answer is 6.
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