Question

if x= √3+1 / √3-1 and y= √3-1 / √3+1 . Find the value of x square+y square+xy

Answer 1

Hello Mate!

$x = \frac{ \sqrt{3} + 1}{ \sqrt{3} - 1 } \times \frac{ \sqrt{3} + 1}{ \sqrt{3} + 1 } \\ \frac{ ({ \sqrt{3 } + 1)}^{2} }{3 - 1} = \frac{4 + 2 \sqrt{3} }{2} \\ = 2 + \sqrt{3}$

$y = \frac{ \sqrt{3} - 1}{ \sqrt{3 } + 1} \times \frac{ \sqrt{3} - 1}{ \sqrt{3} - 1 } \\ = \frac{ {( \sqrt{3} - 1) }^{2} }{3 - 1} \\ = \frac{4 - 2 \sqrt{3} }{2} = 2 - \sqrt{3}$

${x}^{2} = {(2 + \sqrt{3} ) }^{2} \\ = 4 + 3 + 2 \times 2 \times \sqrt{3} \\ = 7 + 4 \sqrt{3}$
${y}^{2} = {(2 - \sqrt{3}) }^{2} \\ = 4 + 3 - 2 \times 2 \times \sqrt{3} \\ = 7 - 4 \sqrt{3}$
$xy = (2 + \sqrt{ 3 } )(2 - \sqrt{3)} \\ = {2}^{2} - { \sqrt{3} }^{2} \\ = 4 - 3 = 1$

${x}^{2} + {y}^{2} + xy = \\ 7 + 4 \sqrt{3} + 7 - 4 \sqrt{3} + 1 \\ 14 + 1 = 15$

Hope it helps☺!

Answer 2

Hey friend,
Here is the answer you were looking for:
$x = \frac{ \sqrt{3} + 1 }{ \sqrt{3} - 1 } \\ \\ on \: rationalizing \: the \: denominator \: we \: get \\ \\ = \frac{ \sqrt{3} + 1 }{ \sqrt{3} - 1} \times \frac{ \sqrt{3} + 1 }{ \sqrt{3} + 1} \\ \\ using \: the \: identity \\ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{( { \sqrt{3}) }^{2} + {(1)}^{2} + 2( \sqrt{3} )(1)}{ {( \sqrt{3}) }^{2} - {(1)}^{2} } \\ \\ = \frac{3 + 1 + 2 \sqrt{3} }{3 - 1} \\ \\ = \frac{4 + 2 \sqrt{3} }{2} \\ \\ = 2 + \sqrt{3} \\ \\ y = \frac{ \sqrt{3} - 1 }{ \sqrt{3} + 1} \\ \\ on \: rationalizing \: the \: denominator \: we \: get \\ \\ = \frac{ \sqrt{3} - 1}{ \sqrt{3} + 1} \times \frac{ \sqrt{3} - 1 }{ \sqrt{3} - 1} \\ \\ using \: the \: identity \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{ {( \sqrt{3}) }^{2} + {(1)}^{2} - 2( \sqrt{3})(1) }{ {( \sqrt{3} )}^{2} - {(1)}^{2} } \\ \\ = \frac{3 + 1 - 2 \sqrt{3} }{3 - 1} \\ \\ = \frac{ 4 - 2 \sqrt{3} }{2} \\ \\ = 2 - \sqrt{3} \\ \\ {x}^{2} + {y}^{2} + xy \\ \\ = {(2 + \sqrt{3} )}^{2} + {(2 - \sqrt{3}) }^{2} + (2 + \sqrt{3} )(2 - \sqrt{3} ) \\ \\ using \: same \: idenities \\ \\ = ( {(2)}^{2} + {( \sqrt{3} )}^{2} + 2(2)( \sqrt{3} ))( {(2)}^{2} + {( \sqrt{3} )}^{2} - 2(2)( \sqrt{3} )) + ( {(2)}^{2} - {( \sqrt{3}) }^{2} ) \\ \\ = (4 + 3 + 4 \sqrt{3} ) + (4 + 3 - 4 \sqrt{3} ) + (4 - 3) \\ \\ = 7 + 4 \sqrt{3} + 7 - 4 \sqrt{3} + 1 \\ \\ = 7 + 7 + 1 \\ \\ = 15$


Hope this helps!!!

@Mahak24

Thanks...
☺☺