Extraneous Solution to a Rational Equation
An extraneous solution to a rational equation is an algebraic solution that causes one or more expressions in the original equation to be undefined. This is a critical concept in Openstax Intermediate Algebra 2E, Chapter 7: Rational Expressions and Functions. Students must always check solutions by substituting back into the original equation and rejecting any that make a denominator zero.
Key Concepts
Property An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined. We note any possible extraneous solutions, $c$, by writing $x \neq c$ next to the equation.
Examples Solve $\frac{x^2}{x 2} = \frac{4}{x 2}$. Multiplying by $x 2$ gives $x^2 = 4$, so $x=2$ or $x= 2$. Since $x=2$ makes the denominator zero, it is an extraneous solution. The only valid solution is $x= 2$.
Solve $\frac{1}{x 4} \frac{1}{x^2 16} = \frac{2}{x+4}$. The LCD is $(x 4)(x+4)$. The equation becomes $(x+4) 1 = 2(x 4)$, which gives $x+3 = 2x 8$, so $x=11$. There are no extraneous solutions.
Common Questions
What is an extraneous solution?
An extraneous solution is a value found algebraically that makes the original equation undefined (such as causing a zero denominator), so it must be rejected.
How do you identify extraneous solutions in rational equations?
After solving, substitute each answer back into the original equation. Reject any value that makes a denominator equal to zero.
Why do extraneous solutions occur in rational equations?
They arise because multiplying both sides by a variable expression can introduce solutions that were not in the original domain.
Where is the concept of extraneous solutions taught in Openstax?
It is covered in Openstax Intermediate Algebra 2E, Chapter 7: Rational Expressions and Functions.
Can a rational equation have no valid solutions?
Yes. If all algebraic solutions are extraneous, the equation has no solution.