$$x^{2}-5x+11=0$$

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 11}}{2}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 11}}{2}$$

$$x=\frac{-\left(-5\right)±\sqrt{25-44}}{2}$$

$$x=\frac{-\left(-5\right)±\sqrt{-19}}{2}$$

$$x=\frac{-\left(-5\right)±\sqrt{19}i}{2}$$

$$x=\frac{5±\sqrt{19}i}{2}$$

$$x=\frac{5+\sqrt{19}i}{2}$$

$$x=\frac{-\sqrt{19}i+5}{2}$$

$$x=\frac{5+\sqrt{19}i}{2}$$ $$x=\frac{-\sqrt{19}i+5}{2}$$

$$x^{2}-5x+11=0$$

$$x^{2}-5x+11-11=-11$$

$$x^{2}-5x=-11$$

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-11+\left(-\frac{5}{2}\right)^{2}$$

$$x^{2}-5x+\frac{25}{4}=-11+\frac{25}{4}$$

$$x^{2}-5x+\frac{25}{4}=-\frac{19}{4}$$

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{19}{4}$$

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{19}{4}}$$

$$x-\frac{5}{2}=\frac{\sqrt{19}i}{2}$$ $$x-\frac{5}{2}=-\frac{\sqrt{19}i}{2}$$

$$x=\frac{5+\sqrt{19}i}{2}$$ $$x=\frac{-\sqrt{19}i+5}{2}$$

$$x ^ 2 -5x +11 = 0$$

$$r + s = 5 $$ $$ rs = 11$$

$$r = \frac{5}{2} - u$$ $$s = \frac{5}{2} + u$$

$$(\frac{5}{2} - u) (\frac{5}{2} + u) = 11$$

$$\frac{25}{4} - u^2 = 11$$

$$-u^2 = 11-\frac{25}{4} = \frac{19}{4}$$

$$u^2 = -\frac{19}{4}$$ $$u = \pm\sqrt{-\frac{19}{4}} = \pm \frac{\sqrt{19}}{2}i $$

$$r =\frac{5}{2} - \frac{\sqrt{19}}{2}i = 2.500 - 2.179i$$ $$s = \frac{5}{2} + \frac{\sqrt{19}}{2}i = 2.500 + 2.179i$$