Solve the equation. Check all proposed solutions. Show work in solving and in checking.
x x 102
---- + ---- = -----
x-3 x+3 x^{2}-9
First let's add the fractions on the left. To add any fraction, you must put them over a common denominator, in this case (x-3)(x+3)
x (x+3) x (x-3)
------ ------ + ------ -------
(x-3) (x+3) (x+3) (x-3)
Multiply out the numerator and denominator then combine like terms
x^{2} + 3x + x^{2} - 3x 2x^{2}
--------------------- = ------
x^{2} - 9 x^{2}-9
Now we can re-state the original problem as:
2x^{2}/(x^{2}-9) = 102/(x^{2}-9)
2x^{2} = 102 (Multiply both sides by (x^{2}-9) )
x^{2} = 51 (Divide both sides by 2)
x = ±√51 (Take the square root of both sides)
To check, plug x = +√51 and x = -√51 into your original equation (below) and make sure the left hand side equals the right hand side. If it does, it's a solution.
2x^{2}/(x^{2}-9) = 102/(x^{2}-9)