解:將方程化為一般形式:$-\frac{2}{3}x^2 + \frac{1}{2}x + 1 = 0,$其中$a = -\frac{2}{3},$$b = \frac{1}{2},$$c = 1。$
判別式$\Delta = b^2 - 4ac = \left(\frac{1}{2}\right)^2 - 4 \times \left(-\frac{2}{3}\right) \times 1 = \frac{1}{4} + \frac{8}{3} = \frac{3 + 32}{12} = \frac{35}{12}。$
由求根公式$x = \frac{-b \pm \sqrt{\Delta}}{2a},$得:
$x = \frac{-\frac{1}{2} \pm \sqrt{\frac{35}{12}}}{2 \times \left(-\frac{2}{3}\right)} = \frac{-\frac{1}{2} \pm \frac{\sqrt{105}}{6}}{-\frac{4}{3}} = \frac{-\frac{3}{6} \pm \frac{\sqrt{105}}{6}}{-\frac{4}{3}} = \frac{-3 \pm \sqrt{105}}{6} \times \left(-\frac{3}{4}\right) = \frac{3 \mp \sqrt{105}}{8},$
即$x_1 = \frac{3 + \sqrt{105}}{8},$$x_2 = \frac{3 - \sqrt{105}}{8}。$