Given (2xsquared +1 )/xsquared minus 2x=k find the values of k for which real roots exist?

(2x^2+1)/(x^2-2x)=k 2x^2+1=k(x^2-2x)

x^2(2-k)+2kx+1=0

for real roots the discriminant must be>0 so 4k^2-8+4k>0 (k+2)(k-1)>0

k>-2 and k>1

so there are real roots for values of k>1

(2x^2 + 1)/(x^2 - 2x) = k

2x^2 + 1 = kx^2 - 2kx

(2 - k)x^2 + 2kx + 1 = 0

Real roots exist when the determinant ≥ 0.

(2k)^2 - 4(2 - k)(1) ≥ 0

4k^2 - 8 + 4k ≥ 0

k^2 + k - 2 ≥ 0

(k + 2)(k - 1) ≥ 0

k ≥ 1 or k ≤ 2