Question

Find the value of $a$ and $b$ if $\frac{\sqrt{3}-1}{\sqrt{3}+1}=a+b\sqrt{3}$

Answer 1

$\frac{\sqrt{3}-1}{\sqrt{3}+1}=a+b\sqrt{3}$

$\Rightarrow$$\frac{(\sqrt{3}-1{)}^2}{(\sqrt{3}-1)(\sqrt{3}+1)}=a+b\sqrt{3}$

$\Rightarrow$$\frac{(\sqrt{3}{)}^2+1^2-2(\sqrt{3}\left)\right(1)}{(\sqrt{3}{)}^2-1^2}=a+b\sqrt{3}$

$\Rightarrow$$\frac{4-2\sqrt{3}}{3-1}=a+b\sqrt{3}$

$\Rightarrow$$2-\sqrt{3}=a+b\sqrt{3}⟹a=2,b=-1$