Find all real values of k for which the equation $\displaystyle x^4-8x^3+kx^2-8x+1$ has at least one repeated root.
First, $\displaystyle x^2[(x+\frac{1}{x})^2-8(x+\frac{1}{x})+(k-2)]=0$ where $\displaystyle x\neq0$
Then $\displaystyle x+\frac{1}{x} = 4 \pm \sqrt(18-k)$ and k= 14, -18 (rejected)
The thing I'm not sure about is whether there are any other real values of k apart from k=14 that would give at least one repeated root...
I'm not sure what you are trying to do here. You seem to be assuming that x must be an integer, which is not true. If x= a, say, is a repeated root then there is a factor of $\displaystyle (x- a)^2$. Try writing $\displaystyle x^4- 8x^3+ kx^2- 8x+ 1= (x- a)^2(x^2+ cx+ d)$.