Sure, let's find the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] given [tex]\( x = 9 - 4 \sqrt{5} \)[/tex].
### Step-by-Step Solution
1. Calculate [tex]\( x \)[/tex]:
- Given [tex]\( x = 9 - 4 \sqrt{5} \)[/tex].
2. Calculate [tex]\( x^2 \)[/tex]:
- We raise [tex]\( x \)[/tex] to the power of 2.
- This gives [tex]\( x^2 = (9 - 4 \sqrt{5})^2 \)[/tex].
- Evaluating this expression, we get [tex]\( x^2 \approx 0.00310562001514181 \)[/tex].
3. Calculate [tex]\( \frac{1}{x^2} \)[/tex]:
- We take the reciprocal of [tex]\( x^2 \)[/tex].
- Therefore, [tex]\( \frac{1}{x^2} \approx \frac{1}{0.00310562001514181} = 321.9968943799899 \)[/tex].
4. Calculate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
- Add [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex].
- [tex]\( x^2 + \frac{1}{x^2} = 0.00310562001514181 + 321.9968943799899 = 322.00000000000506 \)[/tex].
### Final Answer
The value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] is approximately [tex]\( 322.00000000000506 \)[/tex].
