Question

# If frac{3 + √5}{2√5+3} = a + b √5, find the values of a and b.

Solution

( begin{aligned} frac{3+sqrt{5}}{2 sqrt{5}+3} &=frac{(3+sqrt{5})(2 sqrt{5}-3)}{(2 sqrt{5}+3)(2 sqrt{5}-3)} &=frac{6 sqrt{5}-9+10-3 sqrt{5}}{20-9}=frac{3 sqrt{5}+1}{11} &=frac{1}{11}+frac{3}{11} sqrt{5}=a+b sqrt{5} a &=frac{1}{11}, b=frac{3}{11} end{aligned} )