Question

Please solve this problem in a different sheet!! Thank you in advance for your help!!

Answer 1

$Given : a = \frac{ \sqrt{2} - 1 }{ \sqrt{2} + 1 } , b = \frac{ \sqrt{2} + 1 }{ \sqrt{2} - 1 }$

(1).

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

$\frac{ (\sqrt{2} - 1)( \sqrt{2} - 1) }{( \sqrt{2} + 1)( \sqrt{2} - 1) }$

$\frac{( \sqrt{2} - 1)^2 }{( \sqrt{2})^2 - (1)^2 }$

$\frac{( \sqrt{2} - 1)^2 }{1}$

$( \sqrt{2} - 1)^2$

$( \sqrt{2} )^2 - 2 \sqrt{2} + 1$

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

$3 - 2 \sqrt{2}$



(2) 

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

$\frac{( \sqrt{2} + 1)( \sqrt{2} + 1) }{( \sqrt{2} - 1)( \sqrt{2} + 1) }$

$\frac{( \sqrt{2} + 1)^2 }{( \sqrt{2})^2 - 1 }$

$( \sqrt{2} + 1)^2$

$( \sqrt{2} )^2 + 1 * 2 \sqrt{2} + 1$

$2 + 2 \sqrt{2} + 1$

$3 + 2 \sqrt{2}$


Now,

$a^2 + b^2 - 4ab = (3 - 2 \sqrt{2} )^2 + (3 + 2 \sqrt{2} )^2 - 4 (3 - 2 \sqrt{2} )(3 + 2 \sqrt{2} )$

$(17 - 12 \sqrt{2} ) + (12 \sqrt{2} + 17) - (4)$

17 + 17 - 4

30.


Hope this helps!

Answer 2

Hi,

Please see the attached file!


Thanks