$$4x^{2}-5x+8=0$$

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4\times 8}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 4\times 8}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-16\times 8}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-128}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{-103}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{103}i}{2\times 4}$$

$$x=\frac{5±\sqrt{103}i}{2\times 4}$$

$$x=\frac{5±\sqrt{103}i}{8}$$

$$x=\frac{5+\sqrt{103}i}{8}$$

$$x=\frac{-\sqrt{103}i+5}{8}$$

$$x=\frac{5+\sqrt{103}i}{8}$$ $$x=\frac{-\sqrt{103}i+5}{8}$$

$$4x^{2}-5x+8=0$$

$$4x^{2}-5x+8-8=-8$$

$$4x^{2}-5x=-8$$

$$\frac{4x^{2}-5x}{4}=-\frac{8}{4}$$

$$x^{2}-\frac{5}{4}x=-\frac{8}{4}$$

$$x^{2}-\frac{5}{4}x=-2$$

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-2+\left(-\frac{5}{8}\right)^{2}$$

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-2+\frac{25}{64}$$

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{103}{64}$$

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{103}{64}$$

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{103}{64}}$$

$$x-\frac{5}{8}=\frac{\sqrt{103}i}{8}$$ $$x-\frac{5}{8}=-\frac{\sqrt{103}i}{8}$$

$$x=\frac{5+\sqrt{103}i}{8}$$ $$x=\frac{-\sqrt{103}i+5}{8}$$

$$x ^ 2 -\frac{5}{4}x +2 = 0$$

$$r + s = \frac{5}{4} $$ $$ rs = 2$$

$$r = \frac{5}{8} - u$$ $$s = \frac{5}{8} + u$$

$$(\frac{5}{8} - u) (\frac{5}{8} + u) = 2$$

$$\frac{25}{64} - u^2 = 2$$

$$-u^2 = 2-\frac{25}{64} = \frac{103}{64}$$

$$u^2 = -\frac{103}{64}$$ $$u = \pm\sqrt{-\frac{103}{64}} = \pm \frac{\sqrt{103}}{8}i $$

$$r =\frac{5}{8} - \frac{\sqrt{103}}{8}i = 0.625 - 1.269i$$ $$s = \frac{5}{8} + \frac{\sqrt{103}}{8}i = 0.625 + 1.269i$$