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

First let us square both sides:

[sqrt(x+5) + sqrt(x-3)]^2 = 16

==> (x+5) + 2sqrt(x+5)(x-3) + (x-3) = 16

Group similar terms:

==> 2x + 2 + 2sqrt(x^2 + 2x - 15) = 16

divide by 2:

==> x + 1 + sqrt(x^2 + 2x - 15) = 8

Move x + 1 to the right sides:

==> sqrt(x^2 + 2x - 15) = 7x

Now square both sides again:

==> x^2 + 2x - 15 = (7-x)^2

==> x^2 + 2x - 15 = 49-14x + x^2

==> 16x = 64

==> x= 64/16 = 4

==> x= 4

sqrt(x+5)+sqrt(x-3) = 4........(1)

To solve for x.

Multiplying by the conjugate sqrt(x+5) -(x-3) both sides (remembering that (a+b) (a-b) = a^2 - b^2), we get:

x+5 - (x-3) = 4{sqrt(x+5)-sqrt(x-3)}

8 = 4{(sqrt(x+5)-sqrt(x-3)}. Or rewriting as:

sqrt(x+5)-sqrtx-3) = 2. .....................(2).

Eq(1)+eq(2): 2sqrt(x+5) = 4+2= 6

Therefore sqrt(x+5) = 3. Squaring we get x+5 = 9

Or x = 9-5 = 4.

Similarly, Eq(1)-eq(2) gives 2sqrt(x-3) = 4-2 = 2

So sqrt(x-3) = 1. Squaring we get: x-3 = 1

Therefore x = 4

x = 4 is the solution.

To solve for x from sqrt (x+5) + sqrt (x-3) =4 we do the following:

Take the square of both the sides:

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

=>[ x+5 + 2 sqrt (x+5)*sqrt (x-3) + x-3]=16

=> 2x+2+2 sqrt (x+5)*sqrt (x-3) =16

=> x+1 + sqrt (x+5)*sqrt (x-3) =8

=> x +sqrt (x+5)*sqrt (x-3) =7

=> sqrt (x+5)*sqrt (x-3) =7-x

Again square both the sides:

=>(x+5)*(x-3)= 7^2- 14x +x^2

=>x^2+5x-3x-15=49-14x+x^2

=> x^2-x^2+5x-3x+14x-15-49=0

=> 16x-64=0

=>16x=64

=> x=64/16

=>x=4

**Therefore x=4**

We'll multiply, both sides of the equation, by the adjoint expression of the left side.

[sqrt (x+5) + sqrt (x-3)]*[sqrt (x+5) - sqrt (x-3)]= 4*[sqrt (x+5) - sqrt (x-3)]

We'll transform the product from the left side in the difference of squares.

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

(x+5) - (x-3)= 4*[sqrt (x+5) - sqrt (x-3)]

We'll remove the paranthesis from the left side:

x + 5 - x + 3 = 4*[sqrt (x+5) - sqrt (x-3)]

We'll eliminate like terms:

8 = 4*[sqrt (x+5) - sqrt (x-3)]

We'll divide by 4:

2= [sqrt (x+5) - sqrt (x-3)]

We'll add this result to the initial equation:

sqrt(x+5) + sqrt(x-3) + sqrt (x+5) - sqrt (x-3) = 6

We'll eliminate like terms:

2sqrt(x+5) = 6

We'll divide by 2:

sqrt(x+5) = 3

We'll raise to square both sides:

[sqrt(x+5)]^2 = 3^2

x+5= 9

We'll subtract 5 both sides:

x= 9-5

**x= 4**

We'll verify and we'll get x = 4 as valid solution.