what are the two solutions of $2x^2 = -x^2 - 5x - 1$?

- the y-coordinates of the intersection points of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$
- the x-coordinates of the x-intercepts of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$
- the x-coordinates of the intersection points of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$
- the y-coordinates of the y-intercepts of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$

Question

what are the two solutions of $2x^2 = -x^2 - 5x - 1$?

- the y-coordinates of the intersection points of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$
- the x-coordinates of the x-intercepts of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$
- the x-coordinates of the intersection points of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$
- the y-coordinates of the y-intercepts of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Graphical Solution of Equations", "Graphical Intersection of Systems" ], "new_concepts": [], "current_concepts": [ "Graphical Solution of Equations", "Graphical Intersection of Systems" ] } ## Define the system of equations $$ \begin{aligned} y &= 2x^2 \ y &= -x^2 - 5x - 1 \end{aligned} $$ ## Relate algebraic solutions to graphical features $$ \begin{aligned} 2x^2 &= -x^2 - 5x - 1 \ \text{Solutions for } x &\implies \text{x-coordinates of the intersection points} \end{aligned} $$

the y-coordinates of the intersection points of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$

the x-coordinates of the x-intercepts of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$

the x-coordinates of the intersection points of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$

the y-coordinates of the y-intercepts of the graphs of $y = 2x^2$ and $y = -x^2 - 5x - 1$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Graphical Solution of Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Define the system of equations

Relate algebraic solutions to graphical features

Answer: