Question

if x=√3+√2/√3-√2 and y=√3-√2/√3+√2, find the value of (x+y)²
please answer correctly

Answer 1

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

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

and

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

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

$\rm :\longmapsto\: {(x + y)}^{2}$

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

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

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

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

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

Consider,

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

On rationalizing the denominator, we get

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

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

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

$\rm :\longmapsto\:x = \dfrac{5+ 2 \sqrt{6} }{1}$

$\bf\implies \:x = 5 + 2 \sqrt{6}$

Now,

Consider,

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

On rationalizing the denominator, we get

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

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

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

$\rm :\longmapsto\:y = \dfrac{5 - 2 \sqrt{6} }{1}$

$\bf\implies \:y = 5 - 2 \sqrt{6}$

Consider,

$\rm :\longmapsto\: {(x + y)}^{2}$

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

$\: \rm= \: \: {(10)}^{2}$

$\: \rm= \: \: 100$

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)