Prove there are no real roots for m and that it has equal roots?

Question

Prove there are no real roots for m and that it has equal roots:
x^2-5=2m(x-2)

Answer 1

Rewrite this as x² - 2mx + (4m - 5)

The discriminant is 4m² - 4(4m - 5) = 4(m² - 4m + 5). Since the discriminant is > 0 for all m, the quadratic has two real roots, regardless of the value of m.

Answer 2

x^2-2mx +4m-5=0
x=(( 2m+-sqrt( 4m^2-16m+20))/2
which always has real roots for x and different
m=(x^2-5)/2(x-2) which is real for every x not=2.
So nothing of your question makes sense