Question

if X=2-√5/2+√5 , and 2+√5/2-√5 ,find the value of x^2-y^2
​

Answer 1

Step-by-step explanation:

${x}^{2} - {y}^{2} = to \: find$

$x = \frac{2 - \sqrt{5} }{2 + \sqrt{5} } \: and \: y = \frac{2 + \sqrt{5} }{2 - \sqrt{5} }$

$( {x}^{2} - {y}^{2} )$

$= ( { \frac{2 - \sqrt{5} }{2 + \sqrt{5} }) }^{2} - ( {( \frac{2 + \sqrt{5} }{2 - \sqrt{5} } )}^{2}$

$= \frac{4 + 5 - 4 \sqrt{5} }{4 + 5 + 4 \sqrt{5} } - \frac{4 + 5 + 4 \sqrt{5} }{4 + 5 - 4 \sqrt{5} }$

$= \frac{9 - 4 \sqrt{5} }{9 + 4 \sqrt{5} } - ( \frac{9 + 4 \sqrt{5} }{9 - 4 \sqrt{5} } )$

$by \: cross \: multiplication$

$\frac{9(9 - 4 \sqrt{5} ) - 4 \sqrt{5}(9 - 4 \sqrt{5}) \: - (9(9 + 4\sqrt{5}) +4 \sqrt{5}(9 + \sqrt{5} ) }{81 - 80}$

$\frac{81 - 36 \sqrt{5} - 36 \sqrt{5} + 80 - (81 + 36 \sqrt{5} + 36 \sqrt{5} + 180)}{1}$

$161 - 72 \sqrt{5} - (261 + 72 \sqrt{5} )$

$161 - 72 \sqrt{5} - 261 - 72 \sqrt{5}$

$- 422 - 144 \sqrt{5 } \: is \: the \: required \: answer$

u can also use (x-y) (x+y) by using this u get 4root5 OK friend this is urs answer