Question

if p=2-√5/2+√5 and q=2+√5/2-√5 find the value of p + q

Answer 1

$Hey\:there$‼
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$p$ = ( 2 – √5 ) / ( 2 + √5 )

= ( 2 – √5 ) / ( 2 + √5 ) × ( 2 – √5 ) / ( 2 – √5 )

= ( 2 – √5 )² / ( 2² – √5² )

= ( 4 – 4√5 + 5 ) / ( 4 – 5 )

= 9 – 4√5
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$q$ = ( 2 + √5 ) / ( 2 – √5 )

= ( 2 + √5 ) / ( 2 – √5 ) × ( 2 + √5 ) / ( 2 + √5 )

= ( 2 + √5 )² / ( 2² – √5² )

= ( 4 + 4√5 + 5 ) / ( 2 – 5 )

= 9 + 4√5
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Now, putting the values of p and q in the given equation :-

$p$ + $q$ = ( 9 – 4√5 ) + ( 9 + 4√5 )

= 9 – 4√5 + 9 + 4√5

= 18
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Answer 2

Given That
p=(2-√5)/(2+√5)
And q=(2+√5)/(2-√5)
Then We Find The Value Of p+q

$p + q = \frac{2 - \sqrt{5} }{2 + \sqrt{5} } + \frac{2 + \sqrt{5} }{2 - \sqrt{5} }$
$p + q = ( \frac{(2 - \sqrt{5})(2 - \sqrt{5}) + (2 + \sqrt{5} )(2 + \sqrt{5} ) }{(2 + \sqrt{5})(2 - \sqrt{5}) } )$
$p + q = ( \frac{ {(2 - \sqrt{5)} }^{2} + {(2 + \sqrt{5)} }^{2} }{ {2}^{2} - { \sqrt{5} }^{2} } )$
$p + q = ( \frac{4 + 5 - 4( \sqrt{5}) + 4 + 5 + 4( \sqrt{5}) }{4 - 5} )$
$p + q = ( \frac{4 + 5 + 4 + 5}{ - 1})$
$p + q = - 18$