Question

If a = 2 + root 5 / 2 - root 5 and b = 2 - root 5 / 2 + root 5 , then find the value of a^2 - b^2​

Answer 1

Answer :

• - 144\sqrt{5}

Solution :

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In $\sf a = \frac{2 + \sqrt{5}}{2 - \sqrt{5}}$

• $\sf a = \frac{2 + \sqrt{5}}{2 - \sqrt{5}}$

• $\sf a = \frac{2 + \sqrt{5}}{2 - \sqrt{5}} × \frac{2 + \sqrt{5}}{2 + \sqrt{5}}$

• $\sf a = \frac{(2 + \sqrt{5})²}{(2)² - (\sqrt{5})²}$

• $\sf a = \frac{4 + 4\sqrt{5} + 5 }{4 - 5}$

• $\sf b = \frac{4 + 4\sqrt{5} }{- 1 }$

• $\sf a = - 9 + 4\sqrt{5}$

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In $\sf b = \frac{2 - \sqrt{5}}{2 + \sqrt{5}}$

• $\sf b = \frac{2 - \sqrt{5}}{2 + \sqrt{5}}$

• $\sf b = \frac{2 - \sqrt{5}}{2 + \sqrt{5}} × \frac{2 - \sqrt{5}}{2 - \sqrt{5}}$

• $\sf b = \frac{(2 - \sqrt{5})²}{(2)² - ( \sqrt{5})²}$

• $\sf b = \frac{4 - 4\sqrt{5} + 5 }{4 - 5 }$

• $\sf b = \frac{9 - 4\sqrt{5} }{- 1 }$

• $\sf b = - 9 - 4\sqrt{5}$

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Now, As Question,

$\longrightarrow \sf a² - b² = ( - 9 + 4\sqrt{5} )² - ( - 9 - 4\sqrt{5} )²$

$\longrightarrow \sf = ( - 9)² + 2( - 9)(4\sqrt{5}) + ( - 9)² - \bigg((-9)² + 72\sqrt{5} + 81 \bigg)$

$\longrightarrow \sf = 81 - 72\sqrt{5} + 81 - (81 + 72\sqrt{5} + 81 )$

$\longrightarrow \sf = 81 - 75\sqrt{5} + 81 - 81 - 72\sqrt{5} - 81$

$\longrightarrow \sf = \cancel{ 81 } - 75\sqrt{5} \cancel{+ 81} \cancel{- 81 } - 72\sqrt{5} \cancel{- 81}$

$\longrightarrow \sf = \boxed{ - 144 \sqrt{5} }$

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I hope it helps you ❤️✔️