SOLUTION: Solve x/(x+7)-7/(x-7)=x^2+49/x^2-49

Question 421671: Solve
x/(x+7)-7/(x-7)=x^2+49/x^2-49
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Because subtraction of fractions can cause so many problems, I highly recommend that thay be changed to additions of the opposite:

With equations like these I like to start by eliminating the fractions. To eliminate the fractions we multiply both sides of the equation by the Lowest Common Denominator (LCD) of all the denominators. And to find the LCD we need to see the denominators in factored form. So we start by factoring the denominators. The first two don't factor but the last one is a difference of squares:

Looking at the denominators, we can find that the LCD is (x+7)(x-7). Multiplying both sides by the LCD:

On the left side we must use the Distributive Property to multiply:

Now we can cancel:

Note how using the LCD makes all the fractions disappear.

which simplifies as follows:

Now that the fractions are gone we can set about solving for x. This appears to be a quadratic equation but if we subtract from each side, both terms disappear:
-14x + (-49) = 49
Adding 49 we get:
-14x = 98
Dividing by -14 we get:
x = -7

We're not quite done. Whenever you multiply both sides of an equation by an expression that might be zero, you must check your answer(s). (This does not mean it is wrong to multiply by an expression that might be zero!)

When we multiplied both sides of the equation by the LCD, (x+7)(x-7), we multiplied by something that might be zero. In fact, if x = -7 like our solution says, we can see that the LCD was zero. And since a solution found by multiplying both sides by a zero in invalid. We must reject this solution! (Another way to check is to substitute the solution into the original equation:

Checking x = -7:

which simplifies to:

The zero denominators tell us that x = -7 is invalid as a solution. So we must reject it.

Since we rejected the only solution we found, there is no solution to your equation.
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