Question

$x = \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } and \: y = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5 - \sqrt{3} } } find \: the \: value \: of \: 3 {x}^{2} + 3xy + {y}^{2}$
​

Answer 1

$\bf \red x = \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \: and \: \red y = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5 - \sqrt{3} } } \: find \: the \: value \: of \pink{ 3 {x}^{2} + 3xy + {y}^{2} }$

For x

$= \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \\ \\ = \frac{( \sqrt{5}- \sqrt{3}) ^{2} }{( \sqrt{5} ) ^{2} - ( \sqrt{3}) ^{2} } \\ \\ = \frac{ {( \sqrt{5}) }^{2} + {( \sqrt{3} )}^{2} - 2. \sqrt{5} . \sqrt{3} }{5 - 3} \\ \\ = \frac{5 + 3 - 2 \sqrt{15} }{2} \\ \\ = \frac{8 - 2 \sqrt{15} }{2} \\ \\ = \frac{ \cancel2(4 - \sqrt{15} )}{ \cancel2} \\ \\ \bf \red x \: = \boxed{ 4 - \sqrt{15}}$

Now Again,

For y

$= \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ \\ = \frac{( \sqrt{5} + \sqrt{3}) ^{2} }{( \sqrt{5} ) ^{2} - ( \sqrt{3}) ^{2} } \\ \\ = \frac{ {( \sqrt{5}) }^{2} + {( \sqrt{3} )}^{2} + 2. \sqrt{5} . \sqrt{3} }{5 - 3} \\ \\ = \frac{5 + 3 + 2 \sqrt{15} }{2} \\ \\ = \frac{8 + 2 \sqrt{15} }{2} \\ \\ = \frac{ \cancel2(4 + \sqrt{15} )}{ \cancel2} \\ \\ \bf \red y \: = \boxed{ 4 + \sqrt{15}}$

Now According to question -

Putting the values of x = 4 - √15 & y = 4 + √15

⟹ 3x² + 3xy + y²

$\implies \: 3(4 - \sqrt{15} )^{2} + 3(4 - \sqrt{15} )(4 + \sqrt{15} ) + (4 + \sqrt{15})^{2} \\ \\ \implies \: 3( {4}^{2} + { \sqrt{15} }^{2} - 2.4. \sqrt{15}) + 3( {4}^{2} - { \sqrt{15} }^{2} ) + ({4}^{2} + { \sqrt{15}}^{2} + 2.4. \sqrt{15} ) \\ \\ \implies \: 3(16 + 15 - 8 \sqrt{15} ) + 3(16 - 15) + 16 + 15 + 8 \sqrt{15} \\ \\ \implies 48 + \cancel{45} - 24 \sqrt{15} + 48 \cancel{- 45} + 16 + 15 + 8 \sqrt{15} \\ \\ \implies \: 127 - 16 \sqrt{15}$

━━━━━━━━━━━━━━━━━━━━━━━

Note :- Scroll from left to right to view full answer.

Hope this helps you :)

Good night !!

Answer 2

Step-by-step explanation:

$x= \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \: \: , \: \: y= \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \\ x= \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \: \: , \: \: y= \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ x = \frac{ {( \sqrt{5} - \sqrt{3}) }^{2} }{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } \: \: , \: \: y = \frac{ {( \sqrt{5} + \sqrt{3} )}^{2} }{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } \\ x = \frac{5 - 2 \sqrt{15} + 3}{5 - 3} \: \: , \: \: y = \frac{5 + 2 \sqrt{15} + 3}{5 - 3} \\ x = \frac{8 - 2 \sqrt{15} }{2} \: \: , \: \: y = \frac{8 + 2 \sqrt{15} }{2} \\ x = \frac{2(4 - \sqrt{15} )}{2} \: \: , \: \: y = \frac{2(4 + \sqrt{15} )}{2} \\ x = 4 - \sqrt{15} \: \: , \: \: y = 4 + \sqrt{15} \\ \blue{ \underline{put \: the \: values \: in \: e {q}^{n} \: 3 {x}^{2} + 3xy + {y}^{2} } : } \\ = 3 {(4 - \sqrt{15} )}^{2} + 3(4 - \sqrt{15} )(4 + \sqrt{15} ) + {(4 + \sqrt{15}) }^{2} \\ = 3(16 - 8 \sqrt{15} + 15) + 3(16 - 15) + 16 + 8 \sqrt{15} + 15 \\ = 3(31 - 8 \sqrt{15} ) + 3 \times 1 + 31 + 8\sqrt{15} \\ = 93 - 24 \sqrt{15} + 34 +8 \sqrt{15} \\ = 127 - 16 \sqrt{15} \\ \\ \red{\mathfrak{\underline{\large{Hope \: It \: Helps \: You }}}} \\ \green{\mathfrak{\underline{\large{Mark \: Me \: Brainliest}}}}$