Question

If x=(√2+1)/(√2-1) and y=(√2-1)/(√2+1) then find the value of x^2+y^2+xy

Answer 1

rationalize x and Y for easy calculation
$x = \frac{ \sqrt{2} + 1}{ \sqrt{2} - 1 } \times \frac{ \sqrt{2} + 1 }{ \sqrt{2} + 1} \\ x = \frac{2 + 1 + 2 \sqrt{2} }{2 - 1}$
$x = \frac{3 + 2 \sqrt{2} }{1}$
$y = \frac{ \sqrt{2 } - 1 }{ \sqrt{2} + 1 } \times \frac{ \sqrt{2 } - 1}{ \sqrt{2} - 1 } \\ y = \frac{2 + 1 - 2 \sqrt{2} }{2 - 1}$
$y = \frac{3 - 2 \sqrt{2} }{1}$
now put the value of x and Y in final equation
${x}^{2} + {y}^{2} + xy=$
$( {3 + 2 \sqrt{2} )}^{2} + ( {3 - 2 \sqrt{2}) }^{2} + \\ (3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )$
$= 9 + 8 + 12 \sqrt{2} + 9 + 8 - 12 \sqrt{2} + \\ 9 - 8 \\ = 9 + 8 + 9 + 8 + 1$
$= 35$
hope you like the explanation