\-x^2 + x + 6 = 2x + 8\

the equation is represented by the system shown here.

this system intersects in place(s).

zero
one
two

Question

\-x^2 + x + 6 = 2x + 8\

the equation is represented by the system shown here.

this system intersects in   place(s).

zero
one
two

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", "Quadratic Graphs" ], "new_concepts": [ "System of Quadratic and Linear Equations", "Graphical Intersection of Systems" ], "current_concepts": [ "Solving Quadratic Equations", "Quadratic Graphs", "System of Quadratic and Linear Equations", "Graphical Intersection of Systems" ] } ## Rearrange the equation into standard quadratic form $$ \begin{aligned} &-x^2 + x + 6 = 2x + 8 \ &-x^2 - x - 2 = 0 \ &x^2 + x + 2 = 0 \end{aligned} $$ ## Calculate the discriminant to find the number of real solutions $$ \begin{aligned} &D = b^2 - 4ac \ &D = 1^2 - 4(1)(2) = 1 - 8 = -7 \ &D < 0 \end{aligned} $$ ## Determine the number of intersection points $$ \begin{aligned} &\text{Since } D < 0, \text{ there are no real solutions.} \ &\text{Graphically, the parabola } y = -x^2 + x + 6 \text{ and the line } y = 2x + 8 \text{ do not intersect.} \ &\text{Number of intersection places} = 0 \end{aligned} $$ This system intersects in zero place(s). zero

one

two

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "System of Quadratic and Linear Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Rearrange the equation into standard quadratic form

Calculate the discriminant to find the number of real solutions

Determine the number of intersection points

Answer: