Solving the Equation: (x/(x1))^2 + (x/(x+1))^2 = 10/9
This problem involves simplifying and solving a rational equation. Let's break down the steps:
1. Simplify the Equation
 Find a Common Denominator: The least common denominator for the fractions on the lefthand side is (x1)^2(x+1)^2.
 Rewrite the equation:
[(x(x+1))^2 + (x(x1))^2] / [(x1)^2(x+1)^2] = 10/9
 Expand the Squares:
(x^2 + x)^2 + (x^2  x)^2 = 10/9 * (x1)^2(x+1)^2
 Simplify:
2x^4 + 2x^2 = 10/9 (x^4  2x^2 + 1)
2. Solve for x

Multiply Both Sides by 9:
18x^4 + 18x^2 = 10x^4  20x^2 + 10

Rearrange to a Quadratic:
8x^4 + 38x^2  10 = 0

Let y = x^2:
8y^2 + 38y  10 = 0

Solve the Quadratic (using the quadratic formula):
y = [b ± √(b^2  4ac)] / 2a
Where a = 8, b = 38, and c = 10.
Solving this gives us two possible values for y:
 y1 ≈ 0.24
 y2 ≈ 5.24

Substitute back x^2 for y:
 x^2 ≈ 0.24
 x^2 ≈ 5.24

Solve for x:
 x ≈ ±√0.24
 x ≈ ±√(5.24) (This solution is imaginary)
3. Check for Extraneous Solutions
Since the original equation has denominators with variables, we need to make sure our solutions don't make any of the denominators zero. We can see that x = 1 and x = 1 would cause the denominators to be zero. Therefore, these values are extraneous solutions.
4. Final Solution
The solutions to the equation (x/(x1))^2 + (x/(x+1))^2 = 10/9 are:
 x ≈ √0.24
 x ≈ √0.24
Remember to always check for extraneous solutions when dealing with rational equations.