Solve \(\dfrac{2}{x + \sqrt{2 - x^2}} + \dfrac{2}{x - \sqrt{2 - x^2}} = x\)
Solve for x:
2/(x - sqrt(2 - x^2)) + 2/(x + sqrt(2 - x^2)) = x
Bring 2/(x - sqrt(2 - x^2)) + 2/(x + sqrt(2 - x^2)) together using the common denominator (sqrt(2 - x^2) - x) (x + sqrt(2 - x^2)):
-(4 x)/((sqrt(2 - x^2) - x) (x + sqrt(2 - x^2))) = x
Multiply both sides by (sqrt(2 - x^2) - x) (x + sqrt(2 - x^2)):
-4 x = x (sqrt(2 - x^2) - x) (x + sqrt(2 - x^2))
x (sqrt(2 - x^2) - x) (x + sqrt(2 - x^2)) = 2 x - 2 x^3:
-4 x = 2 x - 2 x^3
Subtract 2 x - 2 x^3 from both sides:
2 x^3 - 6 x = 0
Factor x and constant terms from the left hand side:
2 x (x^2 - 3) = 0
Divide both sides by 2:
x (x^2 - 3) = 0
Split into two equations:
x = 0 or x^2 - 3 = 0
Add 3 to both sides:
x = 0 or x^2 = 3
Take the square root of both sides:
x = 0 or x = sqrt(3) or x = -sqrt(3)
Get a commond denominator on the left and we have that
2 [ x - √(2 - x^2) ] + 2 [ x + √(2 - x^2) ]
______________________________ = x
x^2 - (2 - x^2)
4x
________ = x
2x^2 - 2
2x
_______ = x
x^2 - 1
2x = x^3 - x
x^3 - 3x = 0
x ( x^2 - 3) = 0
(x- 0)(x - √3) ( x + √3) = 0
Setting each factor to 0 and solving for x gives the solutions
x =0, x = √3 and x = -√3