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

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

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

$$x=\frac{-\left(-2\right)±\sqrt{4-20\times 3}}{2\times 5}$$

$$x=\frac{-\left(-2\right)±\sqrt{4-60}}{2\times 5}$$

$$x=\frac{-\left(-2\right)±\sqrt{-56}}{2\times 5}$$

$$x=\frac{-\left(-2\right)±2\sqrt{14}i}{2\times 5}$$

$$x=\frac{2±2\sqrt{14}i}{2\times 5}$$

$$x=\frac{2±2\sqrt{14}i}{10}$$

$$x=\frac{2+2\sqrt{14}i}{10}$$

$$x=\frac{1+\sqrt{14}i}{5}$$

$$x=\frac{-2\sqrt{14}i+2}{10}$$

$$x=\frac{-\sqrt{14}i+1}{5}$$

$$x=\frac{1+\sqrt{14}i}{5}$$ $$x=\frac{-\sqrt{14}i+1}{5}$$

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

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

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

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

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

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=-\frac{3}{5}+\left(-\frac{1}{5}\right)^{2}$$

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{3}{5}+\frac{1}{25}$$

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{14}{25}$$

$$\left(x-\frac{1}{5}\right)^{2}=-\frac{14}{25}$$

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{-\frac{14}{25}}$$

$$x-\frac{1}{5}=\frac{\sqrt{14}i}{5}$$ $$x-\frac{1}{5}=-\frac{\sqrt{14}i}{5}$$

$$x=\frac{1+\sqrt{14}i}{5}$$ $$x=\frac{-\sqrt{14}i+1}{5}$$

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

$$r + s = \frac{2}{5} $$ $$ rs = \frac{3}{5}$$

$$r = \frac{1}{5} - u$$ $$s = \frac{1}{5} + u$$

$$(\frac{1}{5} - u) (\frac{1}{5} + u) = \frac{3}{5}$$

$$\frac{1}{25} - u^2 = \frac{3}{5}$$

$$-u^2 = \frac{3}{5}-\frac{1}{25} = \frac{14}{25}$$

$$u^2 = -\frac{14}{25}$$ $$u = \pm\sqrt{-\frac{14}{25}} = \pm \frac{\sqrt{14}}{5}i $$

$$r =\frac{1}{5} - \frac{\sqrt{14}}{5}i = 0.200 - 0.748i$$ $$s = \frac{1}{5} + \frac{\sqrt{14}}{5}i = 0.200 + 0.748i$$