(x+7)^1/2 + (x-1)^1/2 = 4

First we will square both sides:

==> [(x+7)^1/2 + (x-1)^1/2]^2 = 4^2

==> (x+7) + 2(x+7)^1/2 (x-1)^1/2 + (x-1) = 16

==> 2x + 6 + 2sqrt[ (x+7)(x-1)] = 16

Now sbtract 6 from both sides:

==> 2x + 3sqrt(x+7)(x-1) = 10

Move 2x to the right sides:

==> 2sqrt(x+7)(x-1) = 10-2x

Divide by 2:

==> sqrt(x+7)(x-1) = 5- x

Now open brackets:

==> sqrt(x^2 + 6x - 7) = 5-x

Now squere both sides:

==> x^2 + 6x - 7 = (5-x)^2

=> x^2 + 6x - 7 = 25 - 10x + x^2

reduce similars:

==> 6x - 7 = -10x + 25

==> 16x = 32

==> x= 32/16 = 2

**==> x= 2**

Let's multiply the adjoint expression of the left side, to the both sides of the equation.

{[(x+7)^1/2] + [(x-1)^1/2]}X{[(x+7)^1/2] - [(x-1)^1/2]}= 4x{[(x+7)^1/2] - [(x-1)^1/2]}

Multiplying the paranthesis from the left side

[(x+7)^1/2]^2 - [(x+7)^1/2]x[(x-1)^1/2] + [(x+7)^1/2]x[(x-1)^1/2] - [(x-1)^1/2]^2= 4x{[(x+7)^1/2] - [(x-1)^1/2]}

(x+7) - (x-1) = 4x{[(x+7)^1/2] - [(x-1)^1/2]}

Opening the paranthesis from the left side

x + 7 -x +1= 4x{[(x+7)^1/2] - [(x-1)^1/2]}

8 = 4x{[(x+7)^1/2] - [(x-1)^1/2]}

- 2= {[(x+7)^1/2] - [(x-1)^1/2]}

Let's sum up this result with the initial equation

[(x+7)^1/2] - [(x-1)^1/2] + [(x+7)^1/2] + [(x-1)^1/2] = 6

One can note that two terms are the same: [(x+7)^1/2] and the other two term are opposite: [(x-1)^1/2], the last ones being reduced.

2x[(x+7)^1/2]=6

[(x+7)^1/2]=3

[(x+7)^1/2]^2=3^2

x+7=9

x=9-7

x=2

After verifying action, we can conclude that x=2 is the solution of the equation.

To find the solutions for the equation

[(x + 7)^1/2] + [ x - 1)^1/2] = 4

we got about it as follows:

[(x + 7)^1/2] + [ x - 1)^1/2] = 4

square both the sides

=> [(x + 7)] + [ x - 1)] + 2*[(x + 7)^1/2]*[ x - 1)^1/2] = 16

=> 2*[(x + 7)^1/2]*[ x - 1)^1/2] = - 2x - 6 +16 = -2x + 10

=> [(x + 7)^1/2]*[ x - 1)^1/2] = -x + 5

=> => (x+7)*(x-1)= x^2 + 25 - 10x

=> x^2 -7 + 6x = x^2 + 25 - 10x

=> 6x -7 = -10x + 25

=> 16 x = 32

=> x = 32/ 16

=> x = 2

**Therefore x = 2.**

To solve (x+7)^(1/2)+(x-1)^(1/2 = 4.

Square both sodes:

x+7 +2{(x+7)(x-1)}^(1/2) +x-1 = 4^2 = 16

2{(x+7)(x-1)}(1/2) = 16-2x-6 = 10-2x.

Divide by 2:

{(x+7)(x-1)}^(1/2} = 5-x

Square both sides:

(x+7)(x-1) = (5-x)^2.

x^2+6x-7 = 25-10x+x^2

6x+10x = 25+7 = 32

16x= 32

x = 32/16 = 2.