Question

if p = 2-√5/2+√5 and q = 2+√5/2-√5 find p+q​

Answer 1

Solution:

Refer the attachment.

Answer 2

Solution

Given:-

$\tt\implies \: p = \dfrac{2 - \sqrt{5} }{2 + \sqrt{5} } \: \: \: and \: \: q = \dfrac{2 + \sqrt{5} }{2 - \sqrt{5} }$

To Find:-

$\tt \implies \: p + q$

Now We can write as

$\tt \implies \: \dfrac{2 - \sqrt{5} }{2 + \sqrt{5} } + \dfrac{2 + \sqrt{5} }{2 - \sqrt{5} }$

Taking LCM

$\tt \implies \: \dfrac{(2 - \sqrt{5} )(2 - \sqrt{5} ) + (2 + \sqrt{5})(2 + \sqrt{5} )}{(2 + \sqrt{5} )(2 - \sqrt{5})}$

Using This identities

$\tt \implies \: (a + b)^{2} = {a}^{2} + {b}^{2} + 2ab \\ \tt \implies \: (a - b)^{2} = {a}^{2} + {b}^{2} - 2ab$

$\tt \implies \: (a - b)(a + b) = {a}^{2} - {b}^{2}$

$\tt \implies \: \dfrac{(2 - \sqrt{5} )^{2} +(2 + \sqrt{5} )^{2} }{(2 {}^{2} - (\sqrt{5}) ^{2} )}$

$\tt \implies \: \dfrac{4 + 5 - 4 \sqrt{5} + 4 + 5 + 4 \sqrt{5} }{4 - 5}$

$\tt \implies \dfrac{9 + 9}{ - 1}$

$\tt \implies - 18$