Extraneous Solutions
Extraneous solutions is a Grade 7 math concept from Yoshiwara Intermediate Algebra describing solutions that emerge during the solving process but do not satisfy the original equation. They arise most commonly in radical equations, rational equations, and logarithmic equations.
Key Concepts
Property An apparent solution that does not satisfy the original equation is called an extraneous solution . Whenever we multiply an equation by an expression containing the variable, we should check that the solution obtained does not cause any of the fractions to be undefined. An algebraic fraction is undefined for any values of $x$ that make its denominator equal to zero.
Examples Solve $\frac{x}{x 5} = \frac{5}{x 5} + 3$. Multiplying by the LCD, $x 5$, gives $x = 5 + 3(x 5)$, which simplifies to $x = 5 + 3x 15$, or $ 2x = 10$, so $x=5$. Since $x=5$ makes the original denominator zero, it is an extraneous solution.
Solve $\frac{x^2}{x 4} = \frac{16}{x 4}$. Multiplying by $x 4$ gives $x^2=16$, so $x=4$ or $x= 4$. The value $x=4$ is an extraneous solution because it makes the denominator zero. The only valid solution is $x= 4$.
Common Questions
What is an extraneous solution?
An extraneous solution is a value that satisfies an intermediate algebraic step but not the original equation. It must be identified and discarded.
When do extraneous solutions appear?
Extraneous solutions appear when you square both sides (radical equations), multiply by an expression containing the variable (rational equations), or take a logarithm of a variable expression.
How do you identify extraneous solutions?
Substitute each solution back into the original equation. If a solution does not make the original equation true, it is extraneous.
Can a quadratic equation produce extraneous solutions?
Quadratics by themselves typically do not produce extraneous solutions, but they can arise when the quadratic comes from squaring a radical or rational equation.