$$4x^{2}-5x+3=0$$

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4\times 3}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 4\times 3}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-16\times 3}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-48}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{-23}}{2\times 4}$$

$$x=\frac{-\left(-5\right)±\sqrt{23}i}{2\times 4}$$

$$x=\frac{5±\sqrt{23}i}{2\times 4}$$

$$x=\frac{5±\sqrt{23}i}{8}$$

$$x=\frac{5+\sqrt{23}i}{8}$$

$$x=\frac{-\sqrt{23}i+5}{8}$$

$$x=\frac{5+\sqrt{23}i}{8}$$ $$x=\frac{-\sqrt{23}i+5}{8}$$

$$4x^{2}-5x+3=0$$

$$4x^{2}-5x+3-3=-3$$

$$4x^{2}-5x=-3$$

$$\frac{4x^{2}-5x}{4}=-\frac{3}{4}$$

$$x^{2}-\frac{5}{4}x=-\frac{3}{4}$$

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-\frac{3}{4}+\left(-\frac{5}{8}\right)^{2}$$

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{3}{4}+\frac{25}{64}$$

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{23}{64}$$

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{23}{64}$$

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{23}{64}}$$

$$x-\frac{5}{8}=\frac{\sqrt{23}i}{8}$$ $$x-\frac{5}{8}=-\frac{\sqrt{23}i}{8}$$

$$x=\frac{5+\sqrt{23}i}{8}$$ $$x=\frac{-\sqrt{23}i+5}{8}$$

$$x ^ 2 -\frac{5}{4}x +\frac{3}{4} = 0$$

$$r + s = \frac{5}{4} $$ $$ rs = \frac{3}{4}$$

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

$$(\frac{5}{8} - u) (\frac{5}{8} + u) = \frac{3}{4}$$

$$\frac{25}{64} - u^2 = \frac{3}{4}$$

$$-u^2 = \frac{3}{4}-\frac{25}{64} = \frac{23}{64}$$

$$u^2 = -\frac{23}{64}$$ $$u = \pm\sqrt{-\frac{23}{64}} = \pm \frac{\sqrt{23}}{8}i $$

$$r =\frac{5}{8} - \frac{\sqrt{23}}{8}i = 0.625 - 0.599i$$ $$s = \frac{5}{8} + \frac{\sqrt{23}}{8}i = 0.625 + 0.599i$$