Question

Find a and b, if √5-2/√5+2 - √5+2/√5-2 = a+b√5

Answer 1

Hello friends!!

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

First we have to rationalise the denominator.

$\frac{ \sqrt{5} - 2 }{ \sqrt{5} + 2 } \times \frac{ \sqrt{5} - 2}{ \sqrt{5} - 2 } - \frac{ \sqrt{5} + 2 }{ \sqrt{5} - 2} \times \frac{ \sqrt{5} + 2}{ \sqrt{5} + 2} = a + b \sqrt{5}$

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

$\frac{ {( \sqrt{5} - 2)}^{2} }{( \sqrt{5} - 2)( \sqrt{5} + 2) } - \frac{ {( \sqrt{5} + 2)}^{2} }{( \sqrt{5} + 2)( \sqrt{5} + 2) } = a + b \sqrt{5}$

Using identity:

( a - b )² = a² + b² - 2ab

( a + b )² = a² + b² + 2ab

( a - b )( a + b ) = a² - b²

$\frac{ {( \sqrt{5} )}^{2} + {(2)}^{2} - 2 \times 2 \times \sqrt{5} }{ {( \sqrt{5} )}^{2} - {(2)}^{2} } - \frac{ {( \sqrt{5} )}^{2} + {(2)}^{2} + 2 \times 2 \times \sqrt{5} }{ {( \sqrt{5} )}^{2} - {(2)}^{2} } = a + b \sqrt{5}$

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

$\frac{9 - 4 \sqrt{5} }{1} - \frac{9 + 4 \sqrt{5} }{1} = a + b \sqrt{5}$

$9 - 4 \sqrt{5} - (9 + 4\sqrt{5} ) = a + b \sqrt{5}$

$9 - 4 \sqrt{5} - 9 - 4 \sqrt{5} = a + b \sqrt{5}$

$- 8 \sqrt{5} = a + b \sqrt{5}$

Comparing these values,

a = 0

b = - 8

HOPE IT HELPS YOU...

Answer 2

Hey friend,
Here is the answer you were looking for:
$\frac{ \sqrt{5} - 2}{ \sqrt{5} + 2 } - \frac{ \sqrt{5} + 2 }{ \sqrt{5} - 2} = a + b \sqrt{5} \\ \\ on \: rationalizing \: the \: denominator \: we \: get \\ \\ = (\frac{ \sqrt{5} - 2}{ \sqrt{5} + 2} \times \frac{ \sqrt{5} - 2}{ \sqrt{5} - 2}) - ( \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2} \times \frac{ \sqrt{5} + 2 }{ \sqrt{5} + 2} ) \\ \\ using \: the \: identities \\ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = (\frac{ {( \sqrt{5} )}^{2} + {(2)}^{2} - 2 \times \sqrt{5} \times 2}{ {( \sqrt{5} )}^{2} - {(2)}^{2} } )- (\frac{ {( \sqrt{5} })^{2} + {(2)}^{2} + 2 \times \sqrt{5} \times 2 }{ {( \sqrt{5}) }^{2} - {(2)}^{2} } ) \\ \\ = ( \frac{5 + 4 - 4 \sqrt{5} }{5 - 4} ) - ( \frac{5 + 4 + 4 \sqrt{5} }{5 - 4} ) \\ \\ = ( 9 - 4 \sqrt{5} ) - (9 + 4 \sqrt{5} ) \\ \\ = 9 - 4 \sqrt{5} - 9 - 4 \sqrt{5} \\ \\ = - 4 \sqrt{5} - 4 \sqrt{5} \\ \\ - 8 \sqrt{5} = a + b \sqrt{5} \\ \\ a = 1 \\ \\ b = - 8$

Hope this helps!!!!

@Mahak24

Thanks....
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