First we check for bad values. Factoring the left-hand side of the equation yields
which means that -7 and 9 are "bad" values that can't possibly be solutions.
Now we can go ahead and solve this sucker.
Way 1: To eliminate the denominators, multiply both sides of the equation by (x + 7) and both sides of the equation by x – 9. This gives us
x2 – 2x + 81 = x(x – 9) + x(x + 7),
which simplifies to
The solutions to this equation are x = 9 and x = -9. However, checking back to see the "bad" values shows us that x = 9 is an extraneous solution and not a real solution to the original equation. Well, well, well, x = 9. The mask is coming off now, isn't it?
The only solution to the original equation is x = -9.
Way 2: The left-hand side of the equation is already a single fraction. To turn the right-hand side into a single fraction, we need to do some addition. Bust out your scientific calculator. On second thought, though, we may be able to do this one manually.
Now that the denominators of the fractions on either side of the = sign are the same, we need the numerators to be the same. We need to solve the equation
x2 – 2x + 81 = 2x2 – 2x.
This is the same equation we had to solve using Way 1, so we'll find the same answer. If we don't, something may be rotten in the state of Denmark, or whatever state you live in.