for what values of (m) does the graph of (y = 3x^2 + 7x + m) have two (x)-intercepts?

(m > \frac{25}{3})

(m < \frac{25}{3})

(m < \frac{49}{12})

(m > \frac{49}{12})

Question

for what values of (m) does the graph of (y = 3x^2 + 7x + m) have two (x)-intercepts? (m > \frac{25}{3}) (m < \frac{25}{3}) (m < \frac{49}{12}) (m > \frac{49}{12})

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Discriminant of Quadratic Equation", "Nature of Quadratic Roots" ], "new_concepts": [], "current_concepts": [ "Discriminant of Quadratic Equation", "Nature of Quadratic Roots", "Zeros of Quadratic Functions" ] } ## Identify coefficients of the quadratic function $$ \begin{aligned} &y = 3x^2 + 7x + m\ &a = 3,\quad b = 7,\quad c = m \end{aligned} $$ ## Set up the discriminant inequality for two x-intercepts $$ \begin{aligned} &D = b^2 - 4ac > 0\ &7^2 - 4(3)(m) > 0 \end{aligned} $$ ## Solve the inequality for m $$ \begin{aligned} &49 - 12m > 0\ &12m < 49\ &m < \frac{49}{12} \end{aligned} $$

(A) $m > \frac{25}{3}$

(B) $m < \frac{25}{3}$

(C) $m < \frac{49}{12}$

(D) $m > \frac{49}{12}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Discriminant of Quadratic Equation" ] }

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QUESTION IMAGE

Question

Explanation:

Identify coefficients of the quadratic function

Set up the discriminant inequality for two x-intercepts

Solve the inequality for m

Answer: