Question

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Answer Given

$a = \frac{67}{109} \: \: \: \: b = \frac{ - 2}{109}$
I want Explanation Solution​

Answer 1

$\mathsf{Given :\;\dfrac{2 + 3\sqrt{5}}{4 + 5\sqrt{5}}}$

$\mathsf{Multiplying\;numerator\;and\;denominator\;with\;4 - 5\sqrt{5},\;we\;get :}$

$\mathsf{\implies \dfrac{(2 + 3\sqrt{5})(4 - 5\sqrt{5})}{(4 + 5\sqrt{5})(4 - 5\sqrt{5})}}$

★  We know that : (a + b)(a - b) = a² - b²

$\mathsf{\implies \dfrac{(2)(4) - (2)(5\sqrt{5}) + (4)(3\sqrt{5}) - (5\sqrt{5})(3\sqrt{5})}{(4)^2 - (5\sqrt{5})^2}}$

$\mathsf{\implies \dfrac{8 - 10\sqrt{5} + 12\sqrt{5} - (15)(\sqrt{5})^2}{16 - 25(\sqrt{5})^2}}$

$\mathsf{\implies \dfrac{8 + 2\sqrt{5} - (15)(5)}{16 - 25(5)}}$

$\mathsf{\implies \dfrac{8 + 2\sqrt{5} - 75}{16 - 125}}$

$\mathsf{\implies \dfrac{2\sqrt{5} - 67}{-109}}$

$\mathsf{\implies \dfrac{67}{109} - \dfrac{2\sqrt{5}}{109}}$

$\mathsf{\implies \dfrac{67}{109} - \dfrac{2\sqrt{5}}{109} = a + b\sqrt{5}}$

Comparing on both sides, We can notice that :

★  $\mathsf{a = \dfrac{67}{109}}$

★  $\mathsf{b = \dfrac{-2}{109}}$