Question

ANSWER CORRECTLY WITHOUT SPAMING​

Answer 1

Answer:

323 is correct answer

hence it help u

Answer 2

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:x = \dfrac{ \sqrt{5} - 2}{ \sqrt{5} + 2}$

On rationalizing the denominator, we get

$\rm :\longmapsto\:x = \dfrac{ \sqrt{5} - 2}{ \sqrt{5} + 2} \times \dfrac{ \sqrt{5} - 2 }{ \sqrt{5} - 2 }$

We know,

$\boxed{ \rm{ (x + y)(x - y) = {x}^{2} - {y}^{2}}}$

So, using this identity, we get

$\rm :\longmapsto\:x = \dfrac{ {( \sqrt{5} - 2)}^{2} }{ {( \sqrt{5} )}^{2} - {2}^{2} }$

We know,

$\boxed{ \rm{ {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy}}$

So, using this identity, we get

$\rm :\longmapsto\:x = \dfrac{5 + 4 - 2 \times 2 \times \sqrt{5} }{5 - 4}$

$\rm :\longmapsto\:x = \dfrac{9 - 4\sqrt{5} }{1}$

$\bf\implies \:x = 9 - 4 \sqrt{5} - - (1)$

Consider,

$\rm :\longmapsto\:y = \dfrac{ \sqrt{5} + 2}{ \sqrt{5} - 2}$

On rationalizing the denominator, we get

$\rm :\longmapsto\:y = \dfrac{ \sqrt{5} + 2}{ \sqrt{5} - 2} \times \dfrac{ \sqrt{5} + 2}{ \sqrt{5} + 2}$

$\rm :\longmapsto\:y = \dfrac{( \sqrt{5} + 2)^{2} }{( \sqrt{5}) {}^{2} - 2 {}^{2} }$

We know that,

$\boxed{ \rm{ {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}}$

and

$\boxed{ \rm{ (x + y)(x - y) = {x}^{2} - {y}^{2}}}$

So, using these Identities, we get

$\rm :\longmapsto\:y = \dfrac{5 + 4 + 2 \times 2 \times \sqrt{5} }{5 - 4}$

$\rm :\longmapsto\:y = \dfrac{9 + 4\sqrt{5} }{1}$

$\bf\implies \:y = 9 + 4 \sqrt{5} - - - (2)$

Now, Consider,

$\red{\rm :\longmapsto\: {x}^{2} + {y}^{2} + xy}$

can be rewritten as

$\red{\rm \: = \: \: {x}^{2} + {y}^{2} + xy + xy - xy}$

$\red{\rm \: = \: \: {x}^{2} + {y}^{2} +2 xy - xy}$

$\red{\rm \: = \: \: {(x + y)}^{2} - xy}$

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

$\rm \: = \: \: {(18)}^{2} + {9}^{2} - {(4 \sqrt{5} )}^{2}$

$\rm \: = \: \: 324 + 81 - 80$

$\rm \: = \: \: 324 + 1$

$\rm \: = \: \: 325$

Hence,

$\red{\bf :\longmapsto\: {x}^{2} + {y}^{2} + xy = 325}$