Question

x = √3 – √2 / √3 + √2 y = √3 + √2 / √3 – √2 . Find the value of x² – y² .




( don't give unnecessary answers ) ( give correct answer only )​

Answer 1

Answer:

We will first rationalize

again

(a+b)2 =a2 +b2 +2ab put then use

x2 = 49 -20√6

next

same process

y2 =49 +20√6

x2 - y2= 49-20√6 - 49 -20√6

= - 40√6

correct ans

Answer 2

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

$\rm :\longmapsto\:x = \dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} }$

and

$\rm :\longmapsto\:y = \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\: {x}^{2} - {y}^{2}$

$\begin{gathered}\Large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}$

$\green{\boxed{ \bf \: {(x - y)}^{2} \: = {x}^{2} - 2xy + {y}^{2} }}$

$\green{\boxed{ \bf \: {(x + y)}^{2} \: = {x}^{2} + 2xy + {y}^{2} }}$

$\green{\boxed{ \bf \: (x + y)(x - y) \: = \: {x}^{2} - {y}^{2} }}$

$\green{\boxed{ \bf \: {(x + y)}^{2} - {(x - y)}^{2} \: = \: 4xy }}$

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

Given that,

$\green{\bf :\longmapsto\:x = \dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} }}$

$\rm \: = \: \: \dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \times \dfrac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

$\rm \: = \: \: \dfrac{ {( \sqrt{3} - \sqrt{2} )}^{2}}{{( \sqrt{3} )}^{2} - {( \sqrt{2} )}^{2}}$

$\rm \: = \: \: \dfrac{{( \sqrt{3} )}^{2} + {( \sqrt{2} )}^{2} - 2 \times \sqrt{3} \times \sqrt{2} }{3 - 2}$

$\rm \: = \: \: \dfrac{3 + 2 - 2 \sqrt{6} }{1}$

$\rm \: = \: \: 5 - 2 \sqrt{6}$

$\green{ \boxed{\bf\implies \: x = 5 - 2 \sqrt{6}}}$

Consider,

$\green{\bf :\longmapsto\:y = \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }}$

$\rm \: = \: \: \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \times \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} }$

$\rm \: = \: \: \dfrac{ {( \sqrt{3} + \sqrt{2} )}^{2}}{{( \sqrt{3} )}^{2} - {( \sqrt{2} )}^{2}}$

$\rm \: = \: \: \dfrac{{( \sqrt{3} )}^{2} + {( \sqrt{2} )}^{2} + 2 \times \sqrt{3} \times \sqrt{2} }{3 - 2}$

$\rm \: = \: \: \dfrac{3 + 2 + 2 \sqrt{6} }{1}$

$\rm \: = \: \: 5 + 2 \sqrt{6}$

$\green{ \boxed{\bf\implies \: y = 5 + 2 \sqrt{6}}}$

Consider,

$\blue{\bf :\longmapsto\: {x}^{2} - {y}^{2}}$

$\rm \: = \: \: {(5 - 2 \sqrt{6} )}^{2} - {(5 + 2 \sqrt{6} )}^{2}$

$\rm \: = \: \: - \: \bigg( {(5 + 2 \sqrt{6} )}^{2} - {(5 - 2 \sqrt{6}) }^{2} \bigg)$

$\rm \: = \: \: - \: \bigg(4 \times 5 \times 2 \sqrt{6} \bigg)$

$\rm \: = \: \: - \: 40 \sqrt{6}$

$\blue{ \boxed{\bf\implies \:\bf \: {x}^{2} - {y}^{2} = - 40 \sqrt{6}}}$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)