Given:

x = 3 + 2√2

Formula used:

(a2 – b2) = (a – b)(a + b)

Calculation:

x = 3 + 2√2

Squaring on both side

x²= (3 + 2√2)²

x² = 3² + 2 × 3 × 2√2 + (2√2)²

x²= 9 + 12√2 + 8

x²= 17 + 12√2      ----1

To find   \({x^2} + \frac{1}{{{x^2}}}\)

\({x^2} + \frac{1}{{{x^2}}}\; = \;17{\rm{\;}} + {\rm{\;}}12\sqrt 2 + {\rm{\;}}\frac{1}{{17{\rm{\;}} + {\rm{\;}}12\sqrt 2 }}{\rm{\;\;\;\;\;}}\)

By rationalisation method 

⇒ \(17{\rm{\;}} + {\rm{\;}}12\sqrt 2 {\rm{\;}} + {\rm{\;}}\frac{1}{{17{\rm{\;}} + {\rm{\;}}12\sqrt 2 }}\; \times \;\frac{{17{\rm{\;}} - {\rm{\;}}12\sqrt 2 }}{{17{\rm{\;}} - {\rm{\;}}12\sqrt 2 }}\)

Use (a² – b²) = (a – b)(a + b)

⇒ \(17 + {\rm{\;}}12\sqrt {2\;} \; + \;\frac{{17{\rm{\;}} - {\rm{\;}}12\sqrt 2 }}{{{{17}^2}\; - \;{{(12\surd 2)}^2}}}\)

⇒ \(17{\rm{\;}} + {\rm{\;}}12\sqrt {2\;} + \;\frac{{17{\rm{\;}} - {\rm{\;}}12\sqrt {2\;} \;}}{{289\; - \;288}}\)

⇒ 17 + 12√2 + 17 – 12√2

⇒ 34

⇒  \({x^2} + \frac{1}{{{x^2}}} = 34\;\)

∴ The value of \({x^2} + \frac{1}{{{x^2}}}\) is 34