Question

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Answer 1

AnsWer :

- 21√5.

QuestioN :

If $\tt \bullet \: \: \: \: \: \boxed{ \tt a = \dfrac{3 - \sqrt{5} }{3 + \sqrt{5} } } \: \: and \: \: \boxed{\tt b = \dfrac{3 + \sqrt{5} }{3 - \sqrt{5} } }$

SolutioN :

$\tt \bullet \: \: \: \: \: \boxed{ \tt a = \dfrac{3 - \sqrt{5} }{3 + \sqrt{5} } } \: \: and \: \: \boxed{\tt b = \dfrac{3 + \sqrt{5} }{3 - \sqrt{5} } }$

★ Taking a.

$\tt : \implies a = \dfrac{3 - \sqrt{5} }{3 + \sqrt{5} }$

$\tt : \implies a = \dfrac{3 - \sqrt{5} }{3 + \sqrt{5} } \times \dfrac{3 - \sqrt{5} }{3 - \sqrt{5} }$

★ Apply Formula :

• ( a + b )² = a² + b² + 2ab.
• ( a - b )² = a² + b² - 2ab.
• ( a + b )( a - b ) = a² - b².

$\tt : \implies a = \dfrac{ \bigg(3 - \sqrt{5} \bigg)^{2} }{3 ^{2} - \big(\sqrt{5} \big)^{2} }$

$\tt : \implies a = \dfrac{9 + 5 - 6 \sqrt{5} }{9 - 5 }$

$\tt : \implies a = \dfrac{14 - 6 \sqrt{5} }{4 }$

$\tt : \implies a = \dfrac{2(7- 3 \sqrt{5} ) }{4 }$

$\tt : \implies a = \dfrac{7- 3 \sqrt{5} }{2}$

$\rule{200}3$

★ Taking b.

$\tt : \implies b = \dfrac{3 + \sqrt{5} }{3 - \sqrt{5} }$

$\tt : \implies b = \dfrac{3 + \sqrt{5} }{3 - \sqrt{5} } \times \dfrac{3 + \sqrt{5} }{3 + \sqrt{5} }$

$\tt : \implies b = \dfrac{ \bigg(3 + \sqrt{5} \bigg)^{2} }{3^{2} - (\sqrt{5})^{2} }$

$\tt : \implies b = \dfrac{ 9 + 5 + 6 \sqrt{5} }{9 - 5 }$

$\tt : \implies b = \dfrac{ 14 + 6 \sqrt{5} }{9 - 5 }$

$\tt : \implies b = \dfrac{ 14 + 6 \sqrt{5} }{4 }$

$\tt : \implies b = \dfrac{ 2(7+ 3 \sqrt{5} ) }{4 }$

$\tt : \implies b = \dfrac{ 7+ 3 \sqrt{5} }{2 }$

$\rule{200}3$

According to Question, Let's Find

• The value of a² - b².

$\tt \bullet \: \: \: \: \: {a}^{2} - {b}^{2} = (a + b)(a - b)$

$\tt : \implies{a}^{2} - {b}^{2} = \bigg(\dfrac{7- 3 \sqrt{5} }{2} + \dfrac{ 7+ 3 \sqrt{5} }{2 } \bigg)\bigg(\dfrac{7- 3 \sqrt{5} }{2} - \dfrac{ 7+ 3 \sqrt{5} }{2 } \bigg)$

$\tt : \implies{a}^{2} - {b}^{2} = \bigg(\dfrac{7- 3 \sqrt{5} + 7 + 3 \sqrt{5} }{2}\bigg)\bigg(\dfrac{7- 3 \sqrt{5} - (7 + 3 \sqrt{5} ) }{2}\bigg)$

$\tt : \implies{a}^{2} - {b}^{2} = \bigg(\dfrac{7- 3 \sqrt{5} + 7 + 3 \sqrt{5} }{2}\bigg)\bigg(\dfrac{7- 3 \sqrt{5} - 7 - 3 \sqrt{5} ) }{2}\bigg)$

$\tt : \implies{a}^{2} - {b}^{2} = \bigg(\dfrac{14}{2}\bigg)\bigg(\dfrac{- 3 \sqrt{5} - 3 \sqrt{5} }{2}\bigg)$

$\tt : \implies{a}^{2} - {b}^{2} = \bigg(\dfrac{14}{2}\bigg)\bigg(\dfrac{ - 6 \sqrt{5} }{2}\bigg)$

$\tt : \implies{a}^{2} - {b}^{2} = (7)( - 3 \sqrt{5} )$

$\tt : \implies{a}^{2} - {b}^{2} = - 21 \sqrt{5}$

Therefore, the value of a² - b² is - 21√5.