Find the Values
\[ x + \frac{1}{x} = \sqrt{5} \]
Find: \[ x^2 + \frac{1}{x^2} \quad \text{and} \quad x^4 + \frac{1}{x^4} \]
Solution:
Using identity:
\[ \left(x+\frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \]
\[ (\sqrt{5})^2 = x^2 + \frac{1}{x^2} + 2 \]
\[ 5 = x^2 + \frac{1}{x^2} + 2 \]
\[ x^2 + \frac{1}{x^2} = 5 – 2 \]
\[ = 3 \]
Now using identity:
\[ \left(x^2+\frac{1}{x^2}\right)^2 = x^4+\frac{1}{x^4}+2 \]
\[ (3)^2 = x^4+\frac{1}{x^4}+2 \]
\[ 9 = x^4+\frac{1}{x^4}+2 \]
\[ x^4+\frac{1}{x^4} = 9-2 \]
\[ = 7 \]