Question

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

Answer 1

Answer:

0

Step-by-step explanation:

GIVEN:-

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

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

NEED TO FIND:-

${x}^{2} - {y}^{2} + 2xy$

SOLUTION:-

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

We know that,

${\boxed{({a+b})^{2}={a}^{2}+2ab+{b}^{2}}}$

${\boxed{({a-b})^{2}={a}^{2}-2ab+{b}^{2}}}$

${\boxed{(a+b)(a-b)={a}^{2}-{b}^{2}}}$

$=( \frac{4 + 5 - 4 \sqrt{5} }{4 + 5 + 4 \sqrt{5} }) - ( \frac{4 + 5 + 4 \sqrt{5} }{4 + 5 - 4 \sqrt{5} } ) + 2( \frac{4 - 5}{4 - 5} )$

$= \frac{9 - 4 \sqrt{5} }{9 + 4 \sqrt{5} } - \frac{9 + 4 \sqrt{5} }{9 - 4 \sqrt{5} } + 2$

$= \frac{ ( {9 - 4 \sqrt{5} )^{2} -( 9 + 4 \sqrt{5}) }^{2} }{ ({9})^{2} - ({4 \sqrt{5} )}^{2} } +2$

$= \frac{81 + 80 - 8 \sqrt{5} - (81 + 80 - 8 \sqrt{5}) }{81 - 80}+2$

$= \frac{81 + 80 - 8 \sqrt{5} - 81 - 80 + 8 \sqrt{5} }{1} +2$

$= \frac{0}{1}+2$

$= 0+2$

$\color{red}{{\boxed{{x}^{2} - {y}^{2} + 2xy=2}}}$

HOPE THAT HELPS!!