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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Quadratic Equations", "Polynomial Factoring", "AC Method" ], "new_concepts": [], "current_concepts": [ "Solving Quadratic Equations", "Polynomial Factoring", "AC Method" ] } ## Rearrange the quadratic equation Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} 14x^2 &= 23x + 30 \ 14x^2 - 23x - 30 &= 0 \end{aligned} $$ ## Factor the quadratic expression Using the **AC Method** and **Polynomial Factoring** knowledge points $$ \begin{aligned} &a \cdot c = 14 \cdot (-30) = -420 \ &-35 \cdot 12 = -420 \quad \text{and} \quad -35 + 12 = -23 \ &14x^2 - 35x + 12x - 30 = 0 \ &7x(2x - 5) + 6(2x - 5) = 0 \ &(7x + 6)(2x - 5) = 0 \end{aligned} $$ ## Solve for x Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} 7x + 6 = 0 &\implies x = -\frac{6}{7} \ 2x - 5 = 0 &\implies x = \frac{5}{2} \end{aligned} $$ Comparing the solutions, $-\frac{6}{7}$ is the smaller value. Solve the equation by factoring. Enter the smallest solution first. $14x^2 = 23x + 30$ $x =$ $-\frac{6}{7}$, $\frac{5}{2}$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Solving Quadratic Equations" ] }

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Explanation:

Rearrange the quadratic equation

Factor the quadratic expression

Solve for x

Answer: