Question

2-√5/2+3√5=√5 a+b
please answer​

Answer 1

$\sf \: Questions \\ \sf If \frac {2-√5}{2+3√5}=√5 a+b, respectively \: are$

$\qquad \qquad \qquad \sf \: Solution$

We have

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

Now we find values of a and b

$\longmapsto\sf \: \frac{2 - \sqrt{5} }{2 + 3 \sqrt{5} } \\ \\ \\ \longmapsto\sf \: \frac{2 - \sqrt{5} }{2 + 3 \sqrt{5} } \times \frac{2 - \sqrt{5} }{2 + 3 \sqrt{5} } \\ \\ \\ \longmapsto\sf \frac{4 - 6 \sqrt{5} - 2 \sqrt{5} + 15}{4 - 45} \\ \\ \\ \longmapsto\sf \frac{19 - 7 \sqrt{5} }{ - 41} \\ \\ \\ \longmapsto\sf\frac{ - 7 \sqrt{5} + 19 }{ - 41}$

Now comparing with RHS we have

$\sf \implies \: a = \frac{ \cancel - 7}{ \cancel - 41} = \frac{7}{41}\\ \\ \\ \implies \sf b = \frac{19}{ - 41}$