select the correct answer.

what is the solution to the equation?

$\sqrt{3x + 3} - 1 = x$

-1 and 2
-2 and 1
2
1

Question

select the correct answer.

what is the solution to the equation?

$\sqrt{3x + 3} - 1 = x$

-1 and 2
-2 and 1
2
1

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Formula Application", "Viable Solutions in Context" ], "new_concepts": [ "Radical Equations", "Extraneous Solutions" ], "current_concepts": [ "Radical Equations", "Extraneous Solutions", "Quadratic Formula Application", "Viable Solutions in Context" ] } ## Isolate the radical and square both sides $$ \begin{aligned} \sqrt{3x + 3} - 1 &= x \ \sqrt{3x + 3} &= x + 1 \ 3x + 3 &= (x + 1)^2 \ 3x + 3 &= x^2 + 2x + 1 \end{aligned} $$ ## Solve the resulting quadratic equation $$ \begin{aligned} x^2 - x - 2 &= 0 \ (x - 2)(x + 1) &= 0 \ x = 2 \quad &\text{or} \quad x = -1 \end{aligned} $$ ## Verify solutions in the original equation $$ \begin{aligned} \text{For } x = 2: \quad &\sqrt{3(2) + 3} - 1 = \sqrt{9} - 1 = 3 - 1 = 2 \quad (\text{True}) \ \text{For } x = -1: \quad &\sqrt{3(-1) + 3} - 1 = \sqrt{0} - 1 = -1 \quad (\text{True}) \end{aligned} $$

(A) -1 and 2

(B) -2 and 1

(C) 2

(D) 1

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

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QUESTION IMAGE

Question

Explanation:

Isolate the radical and square both sides

Solve the resulting quadratic equation

Verify solutions in the original equation

Answer: