Question

4+2√5/4-3√5 = a+b√5,a, b are rational number, then (a,b)

Answer 1

may be it is right........

Answer 2

Answer:

$a = \frac{ - 61}{29} and \: b = \frac{ - 24}{29}$

Solution:-

We have

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} }$

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5}} \times \frac{4 + 3 \sqrt{5} }{4 + 3 \sqrt{5} }$

$\frac{ {(4 + \sqrt{5})}^{2} }{ {(4)}^{2} - {(3 \sqrt{5}) }^{2} }$

$\frac{ {(4 + 3 \sqrt{5}) }^{2} }{16 - 45}$

$\frac{ {(4 + 3 \sqrt{5}) }^{2} }{ - 29}$

$\frac{ {(4})^{2} + {(3 \sqrt{5}) }^{2} + 2 \times 4 \times 3 \sqrt{5} }{ - 29}$

$\frac{(16 + 45 + 24 \sqrt{5}) }{ - 29}$

$\frac{(61 + 24 \sqrt{5})}{ - 29}$

$( \frac{ - 61}{29}) + (\frac{ - 24}{29} \sqrt{5})$