$$x^2-9\frac{1}{9}x+1=0$$

$$\mathrm{a}=1 ; \mathrm{b}=-9\frac{1}{9} ; \mathrm{c}=1$$

$$x_{1,2}=\frac{-\left(-9{\displaystyle \frac{1}{9}}\right)\pm \sqrt{{\left(-9{\displaystyle \frac{1}{9}}\right)}^2-4·1·1}}{2·1}$$

$$x_{1,2}=\frac{9{\displaystyle \frac{1}{9}}\pm \sqrt{{\left(-{\displaystyle \frac{82}{9}}\right)}^2-4}}{2}$$

$$x_{1,2}=\frac{9{\displaystyle \frac{1}{9}}\pm \sqrt{\left(-{\displaystyle \frac{82}{9}}\right)·{\left(-{\displaystyle \frac{82}{9}}\right)}^{}-4}}{2}$$

$$x_{1,2}=\frac{9{\displaystyle \frac{1}{9}}\pm \sqrt{{{\displaystyle \frac{6724}{81}}}^{}-4}}{2}$$

$$x_{1,2}=\frac{9{\displaystyle \frac{1}{9}}\pm \sqrt{{{\displaystyle \frac{6724}{81}}}^{}-{\displaystyle \frac{324}{81}}}}{2}$$

$$x_{1,2}=\frac{9{\displaystyle \frac{1}{9}}\pm \sqrt{{{\displaystyle \frac{6400}{81}}}^{}}}{2}$$

$$x_{1,2}=\frac{{\displaystyle \frac{82}{9}}\pm {{\displaystyle \frac{80}{9}}}^{}}{2}$$

$$x_1=\frac{{\displaystyle \frac{82}{9}}+{{\displaystyle \frac{80}{9}}}^{}}{2}=\frac{{\displaystyle \frac{162}{9}}}{{\displaystyle \frac{2}{1}}}=\frac{162·1}{9·2}=\frac{162}{18}=9$$

$$x_2=\frac{{\displaystyle \frac{82}{9}}-{\displaystyle \frac{80}{9}}}{2}=\frac{{\displaystyle \frac{2}{9}}}{{\displaystyle \frac{2}{1}}}=\frac{2·1}{9·2}=\frac{2}{18}=\frac{1}{9}$$