To solve for 1/x(x-3) = 2/(x-3) + 3/x, we can simplify and rearrange the terms on the right side:
1/x(x-3) = 2/(x-3) + 3/x
Multiplying both sides by x(x-3):
1 = 2x + 3x(x-3)/x
Expanding the right side:
1 = 2x + 3x^2 - 9x/x
Cancelling x from the right side:
1 = 2x + 3x^2 - 9
Dividing both sides by x^2:
1/x^2 = 2/x + 3 - 9/x^2
Rearranging the terms:
1/x^2 - 9/x^2 = 2/x + 3
Isolating the x terms:
-8/x^2 = 2/x + 3
Multiplying both sides by x^2:
-8 = 2x + 3x^2
Expanding the right side:
-8 = 2x + 3x^2
Rearranging the terms:
3x^2 + 2x + 8 = 0
This is a quadratic equation and can be solved using the quadratic formula or factoring. The solutions are the values of x that make the equation equal to zero.