Question

if x = 2 - root5 divided by 2 + root 5 and y = 2 + root5 divided by 2- root 5 find the value of x square - y square

Answer 1

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Answer 2

Answer:

 The value of $x^{2} - y^{2}$  is  $144\sqrt5$

Step-by-step explanation:

Given,

                      $x = \frac{2-\sqrt5}{2+ \sqrt5}\ and\ y = \frac{2+\sqrt5}{2-\sqrt5}$  

Now,

            $x^{2} - y^{2}$

       $= (\frac{2-\sqrt5}{2+ \sqrt5}\ )^2 - (\frac{2+\sqrt5}{2-\sqrt5})^2$

       $= ( \frac{2-\sqrt5}{2+ \sqrt5}\ + \frac{2+\sqrt5}{2-\sqrt5}) ( \frac{2-\sqrt5}{2+ \sqrt5}\ - \frac{2+\sqrt5}{2-\sqrt5})$

       $= [{\frac{(2-\sqrt5)^2 +(2+\sqrt5)^2}{(2+\sqrt5)(2-\sqrt5)} }][\frac{(2-\sqrt5)^2 -(2+\sqrt5)^2}{(2+\sqrt5)(2-\sqrt5} ]$     [Using, $4ab =(a+b)^2 - (a-b)^2$]

       $=(\frac{18}{-1} )*(\frac{8\sqrt5}{-1} )$                                        [ Using ,$a^{2} - b^{2} = (a+b)(a-b)$]

       $=144\sqrt5$  [Ans]