Question

if X= √2+1/√2-1 and y=√2-1/√2+1,find x²+y²+xy

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Answer 1

$\bf\large\blue{Question\::-}$

If $x = \frac{ \sqrt{2} + 1 }{ \sqrt{2} - 1} \: and \: y \: = \frac{ \sqrt{2 } - 1 }{ \sqrt{2} + 1}$

$\bf\large\blue{Solution\::-}$

Rationalising value of x :-

$\frac{ \sqrt{2} + 1 }{ \sqrt{2} - 1 } \times\frac{ \sqrt{2} + 1 }{ \sqrt{2} + 1 }$

$= \frac{2 + 1 + 2 \sqrt{2} }{2 - 1 }$

$= 3 + 2 \sqrt{2}$

Rationalising value of y :-

We can see y = $\frac{1}{x}$

$= \frac{1}{3 + 2 \sqrt{2} }$

$= \frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }$

$= \frac{3 - 2 \sqrt{2} }{9 - 8}$

$= 3 - 2 \sqrt{2}$

Hence, we got the values of x and y :-

$x = 3 + 2 \sqrt{2}$

$y = 3 - 2 \sqrt{2}$

Now, let's evaluate x² + y² + xy :-

By applying the identity :-

x² + y² + xy = (x + y)² - xy

$= [(3 + 2 \sqrt{2 } )+(3 - 2 \sqrt{2} )] {}^{2} - (\cancel{x }\times \cancel \frac{1}{x} )$

$= (9) {}^{2} - 1$

$= 81 - 1$

$= 80$

$\bf\large\blue{Answer\::-}$

• $\bf{The\:final\: result\:is\:80\:.}$

Answer 2

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