$$\frac { 3 + \sqrt { 7 } } { 3 - \sqrt { 7 } } = a + b \sqrt { 7 }$$
$b=-\frac{\sqrt{7}\left(a-3\sqrt{7}-8\right)}{7}$
$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}=a+b\sqrt{7}$$
$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{3^{2}-\left(\sqrt{7}\right)^{2}}=a+b\sqrt{7}$$
$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{9-7}=a+b\sqrt{7}$$
$$\frac{\left(3+\sqrt{7}\right)\left(3+\sqrt{7}\right)}{2}=a+b\sqrt{7}$$
$$\frac{\left(3+\sqrt{7}\right)^{2}}{2}=a+b\sqrt{7}$$
$$\frac{9+6\sqrt{7}+\left(\sqrt{7}\right)^{2}}{2}=a+b\sqrt{7}$$
$$\frac{9+6\sqrt{7}+7}{2}=a+b\sqrt{7}$$
$$\frac{16+6\sqrt{7}}{2}=a+b\sqrt{7}$$
$$8+3\sqrt{7}=a+b\sqrt{7}$$
$$a+b\sqrt{7}=8+3\sqrt{7}$$
$$b\sqrt{7}=8+3\sqrt{7}-a$$
$$\sqrt{7}b=-a+3\sqrt{7}+8$$
$$\frac{\sqrt{7}b}{\sqrt{7}}=\frac{-a+3\sqrt{7}+8}{\sqrt{7}}$$
$$b=\frac{-a+3\sqrt{7}+8}{\sqrt{7}}$$
$$b=\frac{\sqrt{7}\left(-a+3\sqrt{7}+8\right)}{7}$$
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$a=-\sqrt{7}b+3\sqrt{7}+8$
