solving exponential equations by rewriting the base

solve: $12^{x^2+5x-4} = 12^{2x+6}$

options:
- $x = -5$
- $x = 2$
- $x = 2, x = -5$
- no solution

Question

solving exponential equations by rewriting the base

solve: $12^{x^2+5x-4} = 12^{2x+6}$

options:
- $x = -5$
- $x = 2$
- $x = 2, x = -5$
- no solution

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Solving Exponential Equations", "Quadratic Factoring" ], "current_concepts": [ "Solving Exponential Equations", "Quadratic Factoring" ] } ## Equate the exponents $$ x^2 + 5x - 4 = 2x + 6 $$ ## Rearrange into standard quadratic form $$ \begin{aligned} x^2 + 5x - 2x - 4 - 6 &= 0 \ x^2 + 3x - 10 &= 0 \end{aligned} $$ ## Factor and solve for x $$ \begin{aligned} (x + 5)(x - 2) &= 0 \ x = -5 \quad &\text{or} \quad x = 2 \end{aligned} $$

$x = -5$

$x = 2$

$x = 2, x = -5$

no solution

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

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QUESTION IMAGE

Question

Explanation:

Equate the exponents

Rearrange into standard quadratic form

Factor and solve for x

Answer: