Equation :(

x=1, or

x = 13/2+(3 sqrt(17))/2 (assuming a complex-valued square root)

From: W/A!!.

Solve for x:
8+sqrt(x)-4 x = 10+4 sqrt(x)-5 x

 

Subtract 10+4 sqrt(x)-5 x from both sides:
-2-3 sqrt(x)+x = 0

 

Simplify and substitute y = -3 sqrt(x):
 -2-3 sqrt(x)+x = -2--3 sqrt(x)+((-3 sqrt(x))^2)/(9) = y^2/9-y-2 = 0:
y^2/9-y-2 = 0

 

Multiply both sides by 9:
y^2-9 y-18 = 0

 

Add 18 to both sides:
y^2-9 y = 18

 

Add 81/4 to both sides:
y^2-9 y+81/4 = 153/4

 

Write the left hand side as a square:
(y-9/2)^2 = 153/4

 

Take the square root of both sides:
y-9/2 = (3 sqrt(17))/2 or y-9/2 = -(3 sqrt(17))/2

 

Add 9/2 to both sides:
y = 9/2+(3 sqrt(17))/2 or y-9/2 = -(3 sqrt(17))/2

 

Substitute back for y = -3 sqrt(x):
-3 sqrt(x) = 9/2+(3 sqrt(17))/2 or y-9/2 = -(3 sqrt(17))/2

 

Divide both sides by -3:
sqrt(x) = -3/2-sqrt(17)/2 or y-9/2 = -(3 sqrt(17))/2

 

Raise both sides to the power of two:
x = (-3/2-sqrt(17)/2)^2 or y-9/2 = -(3 sqrt(17))/2

 

Add 9/2 to both sides:
x = (-3/2-sqrt(17)/2)^2 or y = 9/2-(3 sqrt(17))/2

 

Substitute back for y = -3 sqrt(x):
x = (-3/2-sqrt(17)/2)^2 or -3 sqrt(x) = 9/2-(3 sqrt(17))/2

 

Divide both sides by -3:
x = (-3/2-sqrt(17)/2)^2 or sqrt(x) = sqrt(17)/2-3/2

 

Raise both sides to the power of two:
x = (-3/2-sqrt(17)/2)^2 or x = (sqrt(17)/2-3/2)^2

 

8+sqrt(x)-4 x ⇒ 8+sqrt(((-3)/2-(sqrt(17))/(2))^2)-4 (-3/2-sqrt(17)/2)^2 = -11/2 (3+sqrt(17)) ≈ -39.1771
10+4 sqrt(x)-5 x ⇒ 10+4 sqrt(((-3)/2-(sqrt(17))/(2))^2)-5 (-3/2-sqrt(17)/2)^2 = -11/2 (3+sqrt(17)) ≈ -39.1771:
So this solution is correct

 

8+sqrt(x)-4 x ⇒ 8+sqrt(((sqrt(17))/(2)-(3)/2)^2)-4 (sqrt(17)/2-3/2)^2 = 13/2 (sqrt(17)-3) ≈ 7.30019
10+4 sqrt(x)-5 x ⇒ 10+4 sqrt(((sqrt(17))/(2)-(3)/2)^2)-5 (sqrt(17)/2-3/2)^2 = 19/2 (sqrt(17)-3) ≈ 10.6695:
So this solution is incorrect

 

The solution is:
Answer: |x = (-3/2-sqrt(17)/2)^2

Solve for x:
2 sqrt((6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)) = 2+3 sqrt(x)-x

 

Raise both sides to the power of two:
4 (6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x) = (2+3 sqrt(x)-x)^2

 

Subtract (2+3 sqrt(x)-x)^2 from both sides:
4 (6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)-(2+3 sqrt(x)-x)^2 = 0

 

4 (6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)-(2+3 sqrt(x)-x)^2 = 44+44 sqrt(x)-77 x-22 x^(3/2)+11 x^2:
44+44 sqrt(x)-77 x-22 x^(3/2)+11 x^2 = 0

 

Simplify and substitute y = sqrt(x):
 44+44 sqrt(x)-77 x-22 x^(3/2)+11 x^2 = 44+44 sqrt(x)-77 sqrt(x)^2-22 sqrt(x)^3+11 sqrt(x)^4 = 11 y^4-22 y^3-77 y^2+44 y+44 = 0:
11 y^4-22 y^3-77 y^2+44 y+44 = 0

 

The left hand side factors into a product with four terms:
11 (y-1) (y+2) (y^2-3 y-2) = 0

 

Divide both sides by 11:
(y-1) (y+2) (y^2-3 y-2) = 0

 

Split into three equations:
y-1 = 0 or y+2 = 0 or y^2-3 y-2 = 0

 

Add 1 to both sides:
y = 1 or y+2 = 0 or y^2-3 y-2 = 0

 

Substitute back for y = sqrt(x):
sqrt(x) = 1 or y+2 = 0 or y^2-3 y-2 = 0

 

Raise both sides to the power of two:
x = 1 or y+2 = 0 or y^2-3 y-2 = 0

 

Subtract 2 from both sides:
x = 1 or y = -2 or y^2-3 y-2 = 0

 

Substitute back for y = sqrt(x):
x = 1 or sqrt(x) = -2 or y^2-3 y-2 = 0

 

Raise both sides to the power of two:
x = 1 or x = 4 or y^2-3 y-2 = 0

 

Add 2 to both sides:
x = 1 or x = 4 or y^2-3 y = 2

 

Add 9/4 to both sides:
x = 1 or x = 4 or y^2-3 y+9/4 = 17/4

 

Write the left hand side as a square:
x = 1 or x = 4 or (y-3/2)^2 = 17/4

 

Take the square root of both sides:
x = 1 or x = 4 or y-3/2 = sqrt(17)/2 or y-3/2 = -sqrt(17)/2

 

Add 3/2 to both sides:
x = 1 or x = 4 or y = 3/2+sqrt(17)/2 or y-3/2 = -sqrt(17)/2

 

Substitute back for y = sqrt(x):
x = 1 or x = 4 or sqrt(x) = 3/2+sqrt(17)/2 or y-3/2 = -sqrt(17)/2

 

Raise both sides to the power of two:
x = 1 or x = 4 or x = (3/2+sqrt(17)/2)^2 or y-3/2 = -sqrt(17)/2

 

Add 3/2 to both sides:
x = 1 or x = 4 or x = (3/2+sqrt(17)/2)^2 or y = 3/2-sqrt(17)/2

 

Substitute back for y = sqrt(x):
x = 1 or x = 4 or x = (3/2+sqrt(17)/2)^2 or sqrt(x) = 3/2-sqrt(17)/2

 

Raise both sides to the power of two:
x = 1 or x = 4 or x = (3/2+sqrt(17)/2)^2 or x = (3/2-sqrt(17)/2)^2

 

2 sqrt((6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)) ⇒ 2 sqrt((6-2 sqrt(1)-3 1) (2+3 sqrt(1)-1)) = 4
2+3 sqrt(x)-x ⇒ 2+3 sqrt(1)-1 = 4:
So this solution is correct

 

2 sqrt((6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)) ⇒ 2 sqrt((6-2 sqrt(4)-3 4) (2+3 sqrt(4)-4)) = 4 i sqrt(10) ≈ 12.6491 i
2+3 sqrt(x)-x ⇒ 2+3 sqrt(4)-4 = 4:
So this solution is incorrect

 

2 sqrt((6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)) ⇒ 2 sqrt((6-2 sqrt((3/2-sqrt(17)/2)^2)-3 (3/2-sqrt(17)/2)^2) (2+3 sqrt((3/2-sqrt(17)/2)^2)-(3/2-sqrt(17)/2)^2)) = 2 sqrt(273-63 sqrt(17)) ≈ 7.27856
2+3 sqrt(x)-x ⇒ 2+3 sqrt((3/2-sqrt(17)/2)^2)-(3/2-sqrt(17)/2)^2 = 3 (sqrt(17)-3) ≈ 3.36932:
So this solution is incorrect

 

2 sqrt((6-2 sqrt(x)-3 x) (2+3 sqrt(x)-x)) ⇒ 2 sqrt((6-2 sqrt((3/2+sqrt(17)/2)^2)-3 (3/2+sqrt(17)/2)^2) (2+3 sqrt((3/2+sqrt(17)/2)^2)-(3/2+sqrt(17)/2)^2)) = 0
2+3 sqrt(x)-x ⇒ 2+3 sqrt((3/2+sqrt(17)/2)^2)-(3/2+sqrt(17)/2)^2 = 0:
So this solution is correct

 

The solutions are:
Answer: |x = 1                   or                       x = (3/2+sqrt(17)/2)^2