function p has no real solutions.
function r and function p both have no real solutions.
function m and function n both have no real solutions.
function r has no real solutions.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Solving Quadratic Equations", "Parabola Properties" ], "new_concepts": [], "current_concepts": [ "Solving Quadratic Equations", "Parabola Properties", "Complex Roots" ] } ## Identify the x-intercepts of each function $$ \begin{aligned} &\text{Function } n: \text{intersects the } x\text{-axis at } x = -7 \text{ and } x = -3 \implies 2 \text{ real solutions} \ &\text{Function } r: \text{intersects the } x\text{-axis at } x = -8 \text{ and } x = -6 \implies 2 \text{ real solutions} \ &\text{Function } m: \text{intersects the } x\text{-axis at } x = 1 \text{ and } x = 3 \implies 2 \text{ real solutions} \ &\text{Function } p: \text{vertex is at } (2, -3) \text{ and opens downward, never crossing the } x\text{-axis} \implies 0 \text{ real solutions} \end{aligned} $$ ## Evaluate the given options $$ \begin{aligned} &\text{Option 1: "Function } p \text{ has no real solutions."} \implies \text{True} \ &\text{Option 2: "Function } r \text{ and function } p \text{ both have no real solutions."} \implies \text{False (Function } r \text{ has 2 real solutions)} \ &\text{Option 3: "Function } m \text{ and function } n \text{ both have no real solutions."} \implies \text{False (Both have 2 real solutions)} \ &\text{Option 4: "Function } r \text{ has no real solutions."} \implies \text{False (Function } r \text{ has 2 real solutions)} \end{aligned} $$

Function $p$ has no real solutions.

Function $r$ and function $p$ both have no real solutions.

Function $m$ and function $n$ both have no real solutions.

Function $r$ has no real solutions.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Complex Roots" ] }

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Question

Explanation:

Identify the x-intercepts of each function

Evaluate the given options

Answer: