The difference between two roots of the equation $x^3-13x^2+15x+189=0$ is 2, then the roots of the equation are

$\begin{array}{l}\alpha ,\alpha +2,\beta \\ \alpha +\alpha +2+\beta =13\Rightarrow \beta =11-2\alpha \\ \alpha (\alpha +2)+(\alpha +2)\beta +\alpha \beta =15\\ \alpha (\alpha +2)+(\alpha +2)(11-2\alpha )+\alpha (11-2\alpha )=15\\ {\alpha }^2+2\alpha +11\alpha -2{\alpha }^2+22-4\alpha +11\alpha -2{\alpha }^2=15\\ -3{\alpha }^2+20\alpha +7=0\\ 3{\alpha }^2-20\alpha -7=0\\ (\alpha -7)(3\alpha +1)=0\\ \alpha =7,\beta =\frac{-1}{3}\end{array}$

The roots are $\alpha ,\alpha +2,\beta \Rightarrow 7,9,-3$