Question

find rational numbers a and b such that 2-√5/2+√5 = a√5+b​

Answer 1

Hope tjis solution will help you

Answer 2

Answer: a  = 4 , b = -9

Step-by-step explanation:

Given,

$\frac{2-\sqrt{5}}{2+\sqrt{5}}\\\;\\\text{Multiplying Numerator and denominator by } 2-\sqrt{5}\\\;\\=\frac{(2-\sqrt{5})(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})}\\\;\\=\frac{(2\times2)-(2\times\sqrt{5})-(2\times\sqrt{5})+(\sqrt{5}\times\sqrt{5})}{(2^2-(\sqrt{5})^2)}\;\;\;\;\;\;\because(a+b)(a-b)=a^2-b^2\\\;\\=\frac{4-2\sqrt{5}-2\sqrt{5}+5}{4-5}\\\;\\=\frac{9-4\sqrt{5}}{-1}\\\;\\=4\sqrt{5}-9$

Hence,

$4\sqrt{5}-9=a\sqrt{5}+b\\\;\\\text{Comparing both sides}\\\;\\\textbf{a=4\;\;\;and\;\;\;b= -9}$