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

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

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-3\right)}}{2}$$

$$x=\frac{-\left(-3\right)±\sqrt{9+12}}{2}$$

$$x=\frac{-\left(-3\right)±\sqrt{21}}{2}$$

$$x=\frac{3±\sqrt{21}}{2}$$

$$x=\frac{\sqrt{21}+3}{2}$$

$$x=\frac{3-\sqrt{21}}{2}$$

$$x=\frac{\sqrt{21}+3}{2}$$ $$x=\frac{3-\sqrt{21}}{2}$$

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

$$x^{2}-3x-3-\left(-3\right)=-\left(-3\right)$$

$$x^{2}-3x=-\left(-3\right)$$

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

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

$$x^{2}-3x+\frac{9}{4}=3+\frac{9}{4}$$

$$x^{2}-3x+\frac{9}{4}=\frac{21}{4}$$

$$\left(x-\frac{3}{2}\right)^{2}=\frac{21}{4}$$

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

$$x-\frac{3}{2}=\frac{\sqrt{21}}{2}$$ $$x-\frac{3}{2}=-\frac{\sqrt{21}}{2}$$

$$x=\frac{\sqrt{21}+3}{2}$$ $$x=\frac{3-\sqrt{21}}{2}$$

$$x ^ 2 -3x -3 = 0$$

$$r + s = 3 $$ $$ rs = -3$$

$$r = \frac{3}{2} - u$$ $$s = \frac{3}{2} + u$$

$$(\frac{3}{2} - u) (\frac{3}{2} + u) = -3$$

$$\frac{9}{4} - u^2 = -3$$

$$-u^2 = -3-\frac{9}{4} = -\frac{21}{4}$$

$$u^2 = \frac{21}{4}$$ $$u = \pm\sqrt{\frac{21}{4}} = \pm \frac{\sqrt{21}}{2} $$

$$r =\frac{3}{2} - \frac{\sqrt{21}}{2} = -0.791$$ $$s = \frac{3}{2} + \frac{\sqrt{21}}{2} = 3.791$$