$x^2-6x+5=0$

Step 1 ：We are given the quadratic equation $x^2-6x+5=0$.

Step 2 ：This is a quadratic equation in the form of $a{x}^2+bx+c=0$.

Step 3 ：The solutions to a quadratic equation can be found using the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

Step 4 ：In this case, $a=1$, $b=-6$, and $c=5$.

Step 5 ：Substituting these values into the quadratic formula, we get $x=\frac{-(-6)\pm \sqrt{(-6{)}^2-4\ast 1\ast 5}}{2\ast 1}$.

Step 6 ：Solving this, we get $x=\frac{6\pm \sqrt{16}}{2}$.

Step 7 ：This simplifies to $x=\frac{6\pm 4}{2}$.

Step 8 ：So, the solutions to the equation are $x=1$ and $x=5$.

Step 9 ：These are the values of $x$ that satisfy the equation $x^2-6x+5=0$.

Step 10 ：Final Answer: The solutions to the equation are $x=\boxed{{\displaystyle 1}}$ and $x=\boxed{{\displaystyle 5}}$.