$$\frac { 5 + 2 \sqrt { 3 } } { 7 + 4 \sqrt { 3 } } = a - b \sqrt { 3 }$$
$b=\frac{\sqrt{3}\left(a+6\sqrt{3}-11\right)}{3}$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}=a-b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}=a-b\sqrt{3}$$
$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)=a-b\sqrt{3}$$
$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}=a-b\sqrt{3}$$
$$35-6\sqrt{3}-8\times 3=a-b\sqrt{3}$$
$$35-6\sqrt{3}-24=a-b\sqrt{3}$$
$$11-6\sqrt{3}=a-b\sqrt{3}$$
$$a-b\sqrt{3}=11-6\sqrt{3}$$
$$-b\sqrt{3}=11-6\sqrt{3}-a$$
$$\left(-\sqrt{3}\right)b=-a+11-6\sqrt{3}$$
$$\frac{\left(-\sqrt{3}\right)b}{-\sqrt{3}}=\frac{-a+11-6\sqrt{3}}{-\sqrt{3}}$$
$$b=\frac{-a+11-6\sqrt{3}}{-\sqrt{3}}$$
$$b=\frac{\sqrt{3}a}{3}-\frac{11\sqrt{3}}{3}+6$$
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$a=\sqrt{3}b+11-6\sqrt{3}$
