Solve for x square root (x+7) = 4-square root(x-1).

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We'll re-write the expression:

sqrt(x+7) + sqrt(x-1) = 4 (1)

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

[sqrt(x+7) + sqrt(x-1)]*[sqrt(x+7) - sqrt(x-1)]= 4*[sqrt(x+7) - sqrt(x-1)]

We'll have as result of the product of the left side, a difference of squares:

(a-b)(a+b) = a^2 - b^2

We'll put a = sqrt(x+7) and b = sqrt(x-1)

(x+7) - (x-1)= 4*[sqrt(x+7) - sqrt(x-1)]

We'll remove the brackets from the left side:

x + 7 - x +1= 4*[sqrt(x+7) - sqrt(x-1)]

We'll combine and eliminate like terms:

8 = 4*[sqrt(x+7) - sqrt(x-1)]

We'll divide by 4 both sides:

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

We'll add (1)+(2):

sqrt(x+7) + sqrt(x-1) + sqrt(x+7) - sqrt(x-1) = 6

We'll combine and eliminate like terms:

2sqrt(x+7)= 6

We'll divide by 2:

sqrt(x+7)= 3

We'll raise to square both sides:

[sqrt(x+7)]^2= 92= 3^2

x+7= 9

x= 9-7

**x= 2**

**We'll substitute x by 2 and we'll conclude that x= 2 is the solution of the equation.**

We have to find x for sqrt(x+7) = 4 - sqrt(x-1).

Now take the numeric term to one side and bring the other terms to one side.

sqrt(x+7) = 4 - sqrt(x-1)

=>sqrt(x+7) + sqrt(x-1) = 4

finding square of both sides

=> x+7 + x -1 + 2*sqrt(x+7)*sqrt(x-1) = 16

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

simplifying we get

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

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

take the square of both sides again

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

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

cancelling common terms

=> 16x = 32

=> x = 32/ 16

**Therefore x is equal to 32/ 16 =2**

To solve for x in sqrt(x+7) = 4-sqrt(x-1)

We square both sides:

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

x+7 = 4^2 - 2*4*sqrt(x-1)+x-1

7 = 16 -8sqrt(x-1) -1, as the term x on both sides cancels.

7-16 +1 = -8sqrt(x-1)

-8 = -8sqrt(x-1)

Divide by -8:

1 = sqrt (x-1)

We square both sides:

1 = (x-1)

1 +1 = x.

x = 2.

**Therefore x= 2 is the solution.**

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