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

First we will square both sides:

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

Now let us reduce similar terms ( x^2)

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

Now subtract 3x from both sides:

==> -8x + 4 = -2

Now subtract 4 from both sides:

==> -8x = -6

Now we will divide by -8:

==> x = -6/-8

==> x = 6/8 = 3/4

**==> x = 3/4**

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.**

To find x:

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

We square both sides:

x^2 - 5x + 4 = x^2 + 3x -2.

-5x+4 = 3x-2.

-5x-3x = -2-4.

-8x = -6.

-8x/-8 = -6/-8.

x = 3/4.

Therefore x = 3/4 is the solution.