select the correct answer.

at how many points do the graphs of the equations $y = 2x + 5$ and $y = x^2 - 2x - 7$ intersect?

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there is not enough information to determine the number of points of intersection.

Question

select the correct answer.

at how many points do the graphs of the equations $y = 2x + 5$ and $y = x^2 - 2x - 7$ intersect?

0
1
2
there is not enough information to determine the number of points of intersection.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Linear-Quadratic Systems", "Discriminant of a Quadratic" ], "new_concepts": [], "current_concepts": [ "Linear-Quadratic Systems", "Discriminant of a Quadratic" ] } ## Equate the two functions to find intersection points $$ \begin{aligned} y &= 2x + 5 \ y &= x^2 - 2x - 7 \ x^2 - 2x - 7 &= 2x + 5 \end{aligned} $$ ## Rearrange into standard quadratic form $$ \begin{aligned} x^2 - 2x - 2x - 7 - 5 &= 0 \ x^2 - 4x - 12 &= 0 \end{aligned} $$ ## Calculate the discriminant to determine the number of real solutions $$ \begin{aligned} a &= 1,\quad b = -4,\quad c = -12 \ D &= b^2 - 4ac \ D &= (-4)^2 - 4(1)(-12) \ D &= 16 + 48 \ D &= 64 > 0 \end{aligned} $$

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There is not enough information to determine the number of points of intersection.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Linear-Quadratic Systems" ] }

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QUESTION IMAGE

Question

Explanation:

Equate the two functions to find intersection points

Rearrange into standard quadratic form

Calculate the discriminant to determine the number of real solutions

Answer: