Question

If x=2+2√3
Then find x^2-1/x^2

Answer 1

Answer:

$\large{\underline{\boxed{\sf {x}^{2} + \frac{1}{x {}^{2} } = \frac{170 + 63 \sqrt{3} }{8} }}}$

Step-by-step explanation:

Given that ;

x = 2 + 2√3

Now, find the value of 1/x.

$\implies \sf \frac{1}{x} = \frac{1}{2 + 2 \sqrt{3} } \\ \\ \implies \sf \frac{1}{x} = \frac{1}{2 + 2 \sqrt{3} } \times \frac{2 - 2 \sqrt{3} }{2 - 2 \sqrt{3} } \\ \\ \implies \sf \frac{1}{x} = \frac{2 - 2 \sqrt{3} }{(2) {}^{2} - (2 \sqrt{3} ) {}^{2} } \\ \\ \implies \sf \frac{1}{x} = \frac{2 - 2 \sqrt{3} }{4 - 12} \\ \\ \implies \sf \frac{1}{x} = \frac{2(1 - \sqrt{3}) }{ - 8} \\ \\ \implies \sf \frac{1}{x} = \frac{ - (1 - \sqrt{3} )}{ 4} \\ \\ \implies \sf \frac{1}{x} = \frac{ \sqrt{3} - 1 }{4}$

Find the value of x² ;

$\implies \sf {x}^{2} = (2 + 2 \sqrt{3} ) {}^{2} \\ \\ \implies \sf {x}^{2} = (2) {}^{2} + (2 \sqrt{3}) {}^{2} + 2 \times 2 \times 2 \sqrt{3} \\ \\ \implies \sf {x}^{2} = 4 + 12 + 8 \sqrt{3} \\ \\ \implies \sf {x}^{2} = 16 + 8 \sqrt{3}$

Also, find (1/x)² ;

$\implies \sf \frac{1}{x {}^{2} } =( \frac{ \sqrt{3} - 1 }{4} ) {}^{2} \\ \\ \implies \sf \frac{1}{x {}^{2} } = \frac{3 + 1 - 2 \sqrt{3} }{16} \\ \\ \implies \sf \frac{1}{x {}^{2} } = \frac{4 - 2 \sqrt{3} }{16} \\ \\ \implies \sf \frac{1}{x {}^{2} } = \frac{2 - \sqrt{3} }{8}$

Find x² - (1/x)² ;

$\implies \sf {x}^{2} + \frac{1}{x {}^{2} } =16 + 8 \sqrt{3} + \frac{2 - \sqrt{3} }{8} \\ \\ \implies \sf {x}^{2} + \frac{1}{x {}^{2} } = \frac{128 + 64 \sqrt{3} + 2 - \sqrt{3}}{8} \\ \\ \implies \sf {x}^{2} + \frac{1}{x {}^{2} } = \frac{170 + 63 \sqrt{3} }{8}$