$$\frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 } = a + b \sqrt { 3 }$$
$b=-\frac{\sqrt{3}\left(a+\sqrt{3}-2\right)}{3}$
$$\frac{\left(\sqrt{3}-1\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)}=a+b\sqrt{3}$$
$$\frac{\left(\sqrt{3}-1\right)\left(\sqrt{3}-1\right)}{\left(\sqrt{3}\right)^{2}-1^{2}}=a+b\sqrt{3}$$
$$\frac{\left(\sqrt{3}-1\right)\left(\sqrt{3}-1\right)}{3-1}=a+b\sqrt{3}$$
$$\frac{\left(\sqrt{3}-1\right)\left(\sqrt{3}-1\right)}{2}=a+b\sqrt{3}$$
$$\frac{\left(\sqrt{3}-1\right)^{2}}{2}=a+b\sqrt{3}$$
$$\frac{\left(\sqrt{3}\right)^{2}-2\sqrt{3}+1}{2}=a+b\sqrt{3}$$
$$\frac{3-2\sqrt{3}+1}{2}=a+b\sqrt{3}$$
$$\frac{4-2\sqrt{3}}{2}=a+b\sqrt{3}$$
$$2-\sqrt{3}=a+b\sqrt{3}$$
$$a+b\sqrt{3}=2-\sqrt{3}$$
$$b\sqrt{3}=2-\sqrt{3}-a$$
$$\sqrt{3}b=-a+2-\sqrt{3}$$
$$\frac{\sqrt{3}b}{\sqrt{3}}=\frac{-a+2-\sqrt{3}}{\sqrt{3}}$$
$$b=\frac{-a+2-\sqrt{3}}{\sqrt{3}}$$
$$b=\frac{\sqrt{3}\left(-a+2-\sqrt{3}\right)}{3}$$
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$a=-\sqrt{3}b+2-\sqrt{3}$
