Question

(7 + 3√5)/(2 + √5) - (7 - 3√5)/(2 - √5) = a + b√5 find the value of a and b​

Answer 1

Answer:

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Answer 2

Answer:

• a = 0
• b = 2

Explanation:

Given,

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

Taking L. H. S :-

${\implies \: \dfrac{ \big((7 + 3 \sqrt{5} )(2 - \sqrt{5} ) \big) - \big((7 - 3 \sqrt{5}) (2 + \sqrt{5} )\big) }{(2 + \sqrt{5})(2 - \sqrt{5} ) } }$

${ \implies \: \dfrac{ \big( 14 + 6 \sqrt{5} - 7 \sqrt{5} - 15 \big) - \big( 14 + 7 \sqrt{5} - 6 \sqrt{5} - 15\big)}{ {(2)}^{2} - {( \sqrt{5}) }^{2} } }$

${ \implies \: \dfrac{ ( - 1 - 1 \sqrt{5} ) - ( - 1 + 1 \sqrt{5} ) }{ {4} - {{5}}} }$

$\implies \: \dfrac{ - 1 - 1 \sqrt{5} + 1 - 1 \sqrt{5} }{ - 1}$

$\implies \: \dfrac{ - 2 \sqrt{5} }{ - 1}$

$\implies \: {2 \sqrt{5} }$

On comparing with (a + b√5)

$\rm \implies \: 2 \sqrt{5} = a + b \sqrt{5}$

Or we can write it as,

$\rm \implies \: 0 + 2 \sqrt{5} = a + b \sqrt{5}$

Then,

$\rm \implies \: a = 0 \: and \: b = 2$