Question

3-√5/3+2√5=a-b√5, find a and b

Answer 1

Identity used :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$
Now, L.H.S,

$\frac{3 - \sqrt{5} }{3 + 2 \sqrt{5} }$

On rationalizing the denominator we get,


$\frac{3 - \sqrt{5} }{3 + 2 \sqrt{5} } \times \frac{3 - 2 \sqrt{5} }{3 - 2 \sqrt{5} } \\ \\ = \frac{3( 3- 2 \sqrt{5}) - \sqrt{5} (3 - 2 \sqrt{5}) }{ {(3)}^{2} - {(2 \sqrt{5}) }^{2} } \\ \\ = \frac{9 - 6 \sqrt{5} - 3 \sqrt{5} + 10 }{9 - 20} \\ \\ = \frac{19 - 9 \sqrt{5} }{ - 11} \\ \\ = \frac{ - 19 + 9 \sqrt{5} }{11}$

On comparing we get,

$\frac{ - 19 + 9 \sqrt{5} }{11} = a - b \sqrt{5} \\ \\ - \frac{19}{11} + \frac{9}{11} \sqrt{5} = a - b \sqrt{5} \\ \\ a = - \frac{19}{11} \: : \: b = \frac{9}{11}$

Answer 2

here is d sol....
plzz check out the calculation or signs....