Question

[Problema de la olimpiada clasa a 9-a] Rezolvă în R ecuația$\frac{x}{x+1} ^{2} + \frac{x}{x-1} ^{2} = 6$

Answer 1

(x/x+1)^2 + (x/x-1)^2 = 6

x^2/(x+1)^2  + x^2/(x-1)^2 = 6

x^2(x-1)^2 + x^2 (x+1)^2  /  (x+1)^2(x-1)^2  = 6

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

x^2 (x^2-2x+1 + x^2+2x+1) / (x+1)^2(x-1)^2  = 6

x^2 (2x^2 + 2) / (x+1)^2(x-1)^2  = 6   

x^2(2x^2 + 2) = 6 ((x+1)^2(x-1)^2 )       x ≠ +-1

2x^4 + 2x^2 = 6( x^2 - 1)^2

2x^4 + 2x^2 = 6x^4 - 12x^2 + 6

- 4x^4 + 14x^2 - 6 = 0 

-2 (2x^4 - 7x + 3) = 0   / : -2

2x^4 - 7x^2 + 3 = 0     

x^2 = y

2y^2 - 7y + 3 = 0

Δ = 49 - 24 = 25

y1 = (7+5)/4 = 12/4 = 3
y2 = (7 - 5)/4 = 2/4

x1,2 = + - √3
x3,4 = + -√(2/4)