solve the equation $3x = 3\sqrt{2x + 3}$

if there are two solutions, separate them by a comma.

$x =$

Question

solve the equation $3x = 3\sqrt{2x + 3}$

if there are two solutions, separate them by a comma.

$x =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Radical Equations", "Factoring Quadratics", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Radical Equations", "Factoring Quadratics", "Extraneous Solutions" ] } ## Simplify the equation Using the *Radical Equations* knowledge point $$ \begin{aligned} 3x &= 3\sqrt{2x+3} \ x &= \sqrt{2x+3} \end{aligned} $$ ## Square both sides Using the *Radical Equations* knowledge point $$ \begin{aligned} x^2 &= (\sqrt{2x+3})^2 \ x^2 &= 2x + 3 \quad (\text{for } x \ge 0) \end{aligned} $$ ## Solve the quadratic equation Using the *Factoring Quadratics* knowledge point $$ \begin{aligned} x^2 - 2x - 3 &= 0 \ (x - 3)(x + 1) &= 0 \ x = 3 \quad &\text{or} \quad x = -1 \end{aligned} $$ ## Check for extraneous solutions Using the *Extraneous Solutions* knowledge point $$ \begin{aligned} &\text{For } x = 3: \quad 3(3) = 9, \quad 3\sqrt{2(3)+3} = 3\sqrt{9} = 9 \quad (\text{Valid}) \ &\text{For } x = -1: \quad 3(-1) = -3, \quad 3\sqrt{2(-1)+3} = 3\sqrt{1} = 3 \quad (-3 e 3, \text{ Extraneous}) \end{aligned} $$ Solve the equation $3x = 3\sqrt{2x+3}$ If there are two solutions, separate them by a comma. $x =$ 3 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Radical Equations" ] }

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Question

Explanation:

Simplify the equation

Square both sides

Solve the quadratic equation

Check for extraneous solutions

Answer: