Square root of x^2-5x+4 is defined for x = ?

The square root exists if and only if only the expression
x^2-5x+4 is positive or zero.

To determine the range of x
values for the expression x^2-5x+4 to be positive, we'll have to calculate the roots of
the expression
x^2-5x+4.

x1=[-(-5)+sqrt(25-16)]/2

x1=(5+3)/2

x1=4

x2=[-(-5)-sqrt(25-16)]/2

x2=(5-3)/2

x2=1

The
radicand is positive if x belongs to the ranges (-infinite,1] or [4,+infinite) and it is
negative for (1,4).

So, the square root is
defined if x belongs to the reunion of intervals: (-infinite,1] U
[4,+infinite).