what is the solution to the equation $\frac{1}{x} = \frac{x + 3}{2x^2}$?

$x = -3$
$x = -3$ and $x = 0$
$x = 0$ and $x = 3$
$x = 3$

Question

what is the solution to the equation $\frac{1}{x} = \frac{x + 3}{2x^2}$?

$x = -3$
$x = -3$ and $x = 0$
$x = 0$ and $x = 3$
$x = 3$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Equations" ], "new_concepts": [], "current_concepts": [ "Rational Equations", "Denominator Restrictions", "Extraneous Solutions" ] } ## Set up the equation and identify restrictions $$ \frac{1}{x} = \frac{x+3}{2x^2}, \quad x eq 0 $$ ## Cross-multiply and simplify $$ \begin{aligned} 1 \cdot (2x^2) &= x \cdot (x + 3) \ 2x^2 &= x^2 + 3x \ x^2 - 3x &= 0 \end{aligned} $$ ## Solve for x and check restrictions $$ \begin{aligned} x(x - 3) &= 0 \ x = 0 \quad &\text{or} \quad x = 3 \end{aligned} $$ Since $x eq 0$ is a restriction, $x = 0$ is an extraneous solution. Thus, the only valid solution is $x = 3$.

$x = -3$

$x = -3$ and $x = 0$

$x = 0$ and $x = 3$

$x = 3$

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

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QUESTION IMAGE

Question

Explanation:

Set up the equation and identify restrictions

Cross-multiply and simplify

Solve for x and check restrictions

Answer: