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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