List all possible rational roots, use synthetic division to find an actual root, then use this root to solve the equation.
x^3 -7x^2 +13x -7=0
possible rational roots are ±1,±7
division show 1 is a root
x^3 -7x^2 +13x -7 = (x-1)(x^2 - 6x + 7)
so the roots are 1 and 3±√2