Rational Equation: An equation that contains at least one rational expression.
Extraneous Solution: A solution that is derived from the original equation, but it is not actually a solution.
When we end up with an extraneous solution in a rational equation, it is because one of our solutions is an excluded value of one of the rational expressions in the original equation.
Rational Equation
Extraneous Solutions
Rational Expressions
Excluded Values
x^2-3x/x-1=- 2/x-1
x=1
x^2-3x/x-1 and - 2/x-1
x=1
2y/y^2-1=y^2/y+1
y=- 1
2y/y^2-1=y^2/y+1
y=- 1 and y=1
z+3/z^2-4z+4=z-1/z-2+4
None
z+3/z^2-4z+4 and z-1/z-2
z=2
An extraneous solution of a rational equation is always an excluded value of one of the rational expressions in the original equation. However, not every excluded value will be an extraneous solution. It is also worth noting that the extraneous solutions of a rational equation might differ depending on the method we use to solve the equation.
Extra
Extraneous Solutions vs. Method of Solving
Consider the following rational equation.
4x+8/x^2+4x+3=3/x+1
We will solve it using two methods and compare our results.
Using this method, we found x=- 1 and x=1 to be the solutions of our equation. Remember that when dealing with a rational equation, we should always substitute the solutions back into the original equation to check for extraneous solutions.
Substitute
Simplify
Extraneous Solution?
4( - 1)+8/( - 1)^2+4( - 1)+3? =3/- 1+1
4/0? =3/0 *
Yes
4( 1)+8/1^2+4( 1)+3? =3/1+1
12/8=3/2 ✓
No
We can see that only x=1 satisfies the original equation and that x=- 1 is an extraneous solution.
Least Common Denominator
Now we will solve the equation using the least common denominator (LCD). To find the LCD, we will start by factoring the denominator of the rational expression on the left-hand side of the equation.
Let's rewrite our equation using the factored denominator.
4x+8/x^2+4x+3=3/x+1 [0.5em] ⇕ [0.5em] 4x+8/(x+1)(x+3)=3/x+1
The other denominator x+1 cannot be factored and it is a factor of x^2+4x+3. Therefore, the factored version of the denominator on the left-hand side is the LCD.
LCD
x^2+4x+3=(x+1)(x+3)
We are ready to solve our equation. We need to rewrite the right-hand side of the equation to also have the LCD as its denominator. Let's do it!
This time we have only received one solution to the equation. We have already checked that it is not an extraneous solution.
Conclusion
The above example shows that the extraneous solutions depend on the method used. When we solved the equation using the Cross Products Property we obtained one valid solution and one extraneous solution. When we used the LCD, we got only the valid solution.
