sqrt( x^2 - 5x + 4 ) = sqrt( x^2 + 3x -2) find x.

We'll impose the constraint of existence of square
roots:

 x^2 - 5x + 4>=0

We'll
re-write the
inequality:

(x-1)(x-4)>=0

The
intervals of admissible values are (-infinte ,1)U(4 ,
+infinite).

x^2 + 3x -2>=0

x1 =
[-3+sqrt(9+8)]/2

x1 = (-3+sqrt17)/2

x2
= (-3+sqrt17)/2

The intervals of admissible values are (-infinte
,(-3+sqrt17)/2)U((-3+sqrt17)/2 , +infinite).

The solution of the
equation has to belong to the intervals of admissible
values:

(-infinte ,(-3+sqrt17)/2)U(4 ,
+infinite).

Now, we can solve the equation by raising to square both
sides, to get rid of the square roots:

( x^2 - 5x + 4 ) = ( x^2 + 3x
-2)

We'll eliminate and combine like
terms:

-5x - 3x + 4 + 2 = 0

-8x + 6 =
0

-8x = -6

x =
6/8

x = 3/4

We notice
that the value for x doesn't belong to the interval of admissible values for x, so the equation
has no solution.