\frac{a + \frac{1}{2} \cdot \frac{2a\sqrt{7}}{3}}{a - \frac{3}{2} \cdot \frac{2a\sqrt{7}}{3}} = \frac{a + \frac{a\sqrt{7}}{3}}{a - a\sqrt{7}} = \frac{1 + \frac{\sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{\frac{3 + \sqrt{7}}{3}}{1 - \sqrt{7}} = \frac{3 + \sqrt{7}}{3(1 - \sqrt{7})}.