Question

if x = √5 - 2 then find 1) x + 1/x 2) x2 + 1/x2​

Answer 1

Here is your answer :

$x = \sqrt{5} - 2 \\ \\ \frac{1}{x} = \frac{1}{ \sqrt{5} - 2} \\ \\ \frac{1}{x} = \frac{1}{ \sqrt{5} - 2 } \times \frac{ \sqrt{5} + 2 }{ \sqrt{5} + 2} \\ \\ \frac{1}{x} = \frac{ \sqrt{5} + 2} {( \sqrt{5} ) {}^{2} - (2) {}^{2} } \\ \\ \frac{1}{x} = \frac{ \sqrt{5} + 2 }{5 - 4} \\ \\ \frac{1}{x} = \sqrt{5} + 2$

Now,

$x + \frac{1}{x} = \sqrt{5} - 2 + \sqrt{5} + 2 \\ \\ \boxed{ \bf x + \frac{1}{x} = 2 \sqrt{5} } \\ \\ \bf on \: squaring \: both \: side s- \\ \\ (x + \frac{1}{x} ) {}^{2} = (2 \sqrt{5} ) {}^{2} \\ \\ x {}^{2} + \frac{1}{x {}^{2} } + 2 = 20 \\ \\ x {}^{2} + \frac{1}{x {}^{2} } = 20 - 2 \\ \\ \boxed{ \bf x {}^{2} + \frac{1}{x {}^{2} } = 18}$

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