which statement is true about the solution of $\sqrt3{x^2 - 12} = \sqrt3{4x}$?

$x = -2$ is an extraneous solution, and $x = 6$ is a true solution.
$x = 6$ is an extraneous solution, and $x = -2$ is a true solution.
both $x = -2$ and $x = 6$ are extraneous solutions.
both $x = -2$ and $x = 6$ are true solutions.

Question

which statement is true about the solution of $\sqrt3{x^2 - 12} = \sqrt3{4x}$?

$x = -2$ is an extraneous solution, and $x = 6$ is a true solution.
$x = 6$ is an extraneous solution, and $x = -2$ is a true solution.
both $x = -2$ and $x = 6$ are extraneous solutions.
both $x = -2$ and $x = 6$ are true solutions.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Radical Equation Solving", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Radical Equation Solving", "Extraneous Solutions", "Odd Roots" ] } ## Solve the radical equation by cubing both sides $$ \begin{aligned} \sqrt[3]{x^2 - 12} &= \sqrt[3]{4x} \ x^2 - 12 &= 4x \ x^2 - 4x - 12 &= 0 \ (x - 6)(x + 2) &= 0 \ x = 6 \quad &\text{or} \quad x = -2 \end{aligned} $$ ## Check for extraneous solutions in the original equation $$ \begin{aligned} \text{For } x = 6: \quad \sqrt[3]{6^2 - 12} &= \sqrt[3]{24}, \quad \sqrt[3]{4(6)} = \sqrt[3]{24} \quad (\text{True}) \ \text{For } x = -2: \quad \sqrt[3]{(-2)^2 - 12} &= \sqrt[3]{-8} = -2, \quad \sqrt[3]{4(-2)} = \sqrt[3]{-8} = -2 \quad (\text{True}) \end{aligned} $$ ## Identify the correct statement $$ \text{Both } x = -2 \text{ and } x = 6 \text{ are true solutions.} $$

x = -2 is an extraneous solution, and x = 6 is a true solution.

x = 6 is an extraneous solution, and x = -2 is a true solution.

Both x = -2 and x = 6 are extraneous solutions.

Both x = -2 and x = 6 are true solutions.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Radical Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Solve the radical equation by cubing both sides

Check for extraneous solutions in the original equation

Identify the correct statement

Answer: