Sure! Let's solve this step by step.
We start with the given equation:
[tex]\[ x + \frac{1}{x} = 9 \][/tex]
To find the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex], let's square both sides of the equation.
Squaring both sides:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = 9^2 \][/tex]
Expanding the left-hand side using the binomial theorem:
[tex]\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 81 \][/tex]
Simplifying:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 81 \][/tex]
We need to isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]. Subtract 2 from both sides:
[tex]\[ x^2 + \frac{1}{x^2} = 81 - 2 \][/tex]
Simplifying the right-hand side:
[tex]\[ x^2 + \frac{1}{x^2} = 79 \][/tex]
So, the value is:
[tex]\[ x^2 + \frac{1}{x^2} = 79 \][/tex]
