Example Card: Identifying Extraneous Solutions
Identify extraneous solutions in rational equations by checking all answers in the original equation and rejecting values that cause division by zero in Grade 9 Algebra.
Key Concepts
Squaring both sides of an equation can sometimes create a “solution” that is an imposter. This example demonstrates another key idea: identifying extraneous solutions.
Example Problem Solve the equation $\sqrt{x 2} = x 4$.
Step by Step 1. The radical expression is already isolated. Square both sides to eliminate the radical. $$ (\sqrt{x 2})^2 = (x 4)^2 $$ 2. Simplify both sides. Remember to expand the binomial $(x 4)^2$ as a perfect square trinomial. $$ x 2 = x^2 8x + 16 $$ 3. This is a quadratic equation. Set the equation to zero by moving all terms to one side. $$ 0 = x^2 9x + 18 $$ 4. Factor the quadratic expression. $$ 0 = (x 3)(x 6) $$ 5. Use the Zero Product Property to find the two potential solutions. $$ x 3 = 0 \quad \text{or} \quad x 6=0 $$ $$ x = 3 \quad \text{or} \quad x=6 $$ 6. Crucially, check both potential solutions in the original equation to see if they are valid. 7. Check $x=3$: $$ \begin{align } \sqrt{3 2} &\stackrel{?}{=} 3 4 \\ \sqrt{1} &\stackrel{?}{=} 1 \\ 1 &\neq 1 \quad \times \end{align } $$ 8. Check $x=6$: $$ \begin{align } \sqrt{6 2} &\stackrel{?}{=} 6 4 \\ \sqrt{4} &\stackrel{?}{=} 2 \\ 2 &= 2 \quad \checkmark \end{align } $$ 9. The solution $x=3$ is extraneous. The only valid solution is $x=6$.
Common Questions
What is Example Card: Identifying Extraneous Solutions?
Example Card: Identifying Extraneous Solutions is a key concept in Grade 9 math. It involves applying specific rules and properties to simplify expressions, solve equations, or analyze mathematical relationships. Understanding this topic builds foundational skills needed for higher-level algebra and beyond.
How is Example Card: Identifying Extraneous Solutions used in real-world applications?
Example Card: Identifying Extraneous Solutions appears in practical contexts such as financial calculations, engineering problems, and data analysis. Mastering this skill helps students model and solve problems they will encounter in science, technology, and everyday decision-making situations.
What are common mistakes when working with Example Card: Identifying Extraneous Solutions?
Common errors include forgetting to apply rules to all terms, sign errors when working with negatives, and skipping verification steps. Always double-check by substituting answers back into the original problem and reviewing each algebraic step carefully.