Sure! Let's go through the problem step-by-step to show that [tex]\( x^2 + \frac{1}{x^2} = 11 \)[/tex] given that [tex]\( x - \frac{1}{x} = 3 \)[/tex].
1. Start with the given equation:
[tex]\[
x - \frac{1}{x} = 3
\][/tex]
2. Square both sides of the equation to isolate [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex].
[tex]\[
\left( x - \frac{1}{x} \right)^2 = 3^2
\][/tex]
3. Simplify both sides:
The left-hand side is the square of a binomial:
[tex]\[
\left( x - \frac{1}{x} \right)^2 = x^2 - 2 \left( x \cdot \frac{1}{x} \right) + \left( \frac{1}{x} \right)^2
\][/tex]
Using [tex]\( \left( x \cdot \frac{1}{x} = 1 \)[/tex] \), this becomes:
[tex]\[
x^2 - 2 \cdot 1 + \frac{1}{x^2} = x^2 - 2 + \frac{1}{x^2}
\][/tex]
The right-hand side simplifies to:
[tex]\[
3^2 = 9
\][/tex]
4. Substitute back into the equation:
[tex]\[
x^2 - 2 + \frac{1}{x^2} = 9
\][/tex]
5. Add 2 to both sides of the equation to solve for [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[
x^2 + \frac{1}{x^2} - 2 + 2 = 9 + 2
\][/tex]
Simplifying gives us:
[tex]\[
x^2 + \frac{1}{x^2} = 11
\][/tex]
Thus, we have shown that:
[tex]\[
x^2 + \frac{1}{x^2} = 11
\][/tex]
This completes the proof.
