Rational Equations and Extraneous Solutions
A rational equation is an equation which contains at least one rational expression. Recall that a rational expression can also be referred to as an algebraic fraction. In other words, one of the terms has a variable in the denominator.
There is a very important implication when a variable is in the denominator of an expression or equation. Dividing any quantity by zero leads to an undefined value, so whenever the denominator is equal to zero, the equation is undefined. Sometimes, when solving rational equations, we can make all the right moves algebraically, but the solution we get for our variable makes our denominator equal zero. These solutions are known as extraneous solutions.
When solving rational equations, always check to make sure the solution does not make the denominator equal zero. If it does, the solution is extraneous, and should not be included in your final answer as a solution to the rational equation.
A useful strategy in solving rational equations is to rewrite the equation so that every term has a common denominator. The reason for this is because once all terms are written to have a common denominator, we can create an equation with just the numerators. Here is an example:
Because each term in the rational equation has the same expression for its denominator, this also represents a solution to the rational equation. The only thing we need to do is check to make sure that when x = –7, we never have a denominator of zero in any of the terms. Since the denominator is x – 1, and –7 – 1 is non-zero, we know that x = –7 is the solution to this rational equation.
Finding the Least Common Denominator
Now that we have a useful strategy for solving rational functions, let's talk about the processes for creating terms with common denominators.
Let's apply these steps to find the lowest common denominator between these algebraic fractions:
The first step is to factorize each denominator. This means we factor out common factors between terms, if possible, or break numbers down into prime numbers (for example, 4 can be written as 2 • 2)
Next, we look at each factor in the factorizations, and determine the greatest number of times it appears within a single factorization:
We multiply these quantities together to get the common denominator:
Solving Rational Equations using a Common Denominator
Let's solve the following rational equation using our common denominator strategy:
First, we need to find the least common denominator. Luckily for us, each denominator is already factorized, so the least common factor is the product of the 3 denominators:
The tricky part is now adjusting our numerators to the original fractions, in order to be equivalent with a new denominator. To do this, let's express each part of the fraction in factored form. This will help identify which factors need to be attached to the numerator:
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Now we can ignore the denominator (the expression we worked so hard to find!) and create an equation with no denominator at all:
This is actually a quadratic equation we can solve using the quadratic formula, once we set one side of the equation equal to zero:
The quadratic formula is
Be sure to plug in 2.5 and 6 into all denominators from our original rational equation. If either 2.5 or 6 make one of the denominators equal zero, we have an extraneous solution.
Since no denominator equals 0 at x = 2.5, or x = 6, there are no extraneous solutions to the rational equation.
an equation with at least one rational expression