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To solve this we need to get eliminate the fractions. To eliminate the fractions we want to know the Lowest Common Denominator (LCD). To find the LCD we need to know the factors of the all the denominators. So we will start by factoring the denominators.
The first denominator is a trinomial which factors fairly easily. The other two denominators do not factor:
The LCD of these denominators is the product of all the different factors. If a factor appears in more than one denominator, use it just once with the highest exponent is has in these denominators. Your LCD, there fore, is .
To eliminate the fractions we multiply both sides of the equation by the LCD:
On the right side, we need to use the Distributive Property:
Now the fractions disappear because all the denominators cancel out:
leaving:
which simplifies to:
We now have a quadratic equation. To solve it we get one side equal to zero:
... factor ir (or use the quadratic formula):
... and use the Zero Product Property (which tells us that a product is zero only if one of the factors is zero): or
Solving these we get: or
In a problem like this it is more than just a good idea to check our answers. We need to check the answers because when we multiplied both sides by the LCD, we multiplied by an expression that might be zero.
Checking x=6:
As we can see, x=6 makes two of the denominators zero! We must reject x = 6.
Checking x = -1:
No zero denominators so this solution looks good. Check!
So the only solution to the original equation is x = -1.
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