Question

solve 3-✓5/3-2✓5 = a✓5 + b. Find the value of a and b.

Answer 1

Solution is in the attachment.....

Answer 2

Identity used :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$


$\frac{3 - \sqrt{5} }{3 - 2 \sqrt{5} } = a \sqrt{5} + b$

L.H.S.

On rationalizing the denominator we get,

$= \frac{3 - \sqrt{5} }{3 - 2 \sqrt{5} } \times \frac{3 + 2 \sqrt{5} }{3 + 2 \sqrt{5} } \\ \\ = \frac{3(3 + 2 \sqrt{5} ) - \sqrt{5} (3 + 2 \sqrt{5}) }{ {(3)}^{2} - {(2 \sqrt{5} )}^{2} } \\ \\ = \frac{9 + 6 \sqrt{5} - 3 \sqrt{5} - 10 }{9 - 20} \\ \\ = \frac{ - 1 + 3 \sqrt{5} }{ - 11} \\ \\ = \frac{1 - 3 \sqrt{5} }{11} \\ \\ = \frac{ - 3 \sqrt{5} + 1 }{11}$

On comparing L.H.S and R.H.S we get,

$\frac{ - 3\sqrt{5} + 1 }{11} = a \sqrt{5} + b \\ \\ a = \frac{ - 3}{11} \: : \: b = \frac{1}{11}$