both sides of an equation have quadratic expressions. the graph of one quadratic opens upward. the graph of the other opens downward.

what are the possible number of intersections for this system?

0
1
2
infinite

Question

both sides of an equation have quadratic expressions. the graph of one quadratic opens upward. the graph of the other opens downward.

what are the possible number of intersections for this system?

0
  1
  2
  infinite

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Quadratic Functions", "Zeros of Quadratic Functions", "Nature of Quadratic Roots" ], "new_concepts": [ "Intersection of Quadratic Graphs" ], "current_concepts": [ "Quadratic Functions", "Zeros of Quadratic Functions", "Nature of Quadratic Roots", "Intersection of Quadratic Graphs" ] } ## Formulate the system of quadratic equations $$ \begin{aligned} y_1 &= a_1 x^2 + b_1 x + c_1, \quad a_1 > 0 \ y_2 &= a_2 x^2 + b_2 x + c_2, \quad a_2 < 0 \end{aligned} $$ ## Set up the intersection equation $$ \begin{aligned} a_1 x^2 + b_1 x + c_1 &= a_2 x^2 + b_2 x + c_2 \ (a_1 - a_2) x^2 + (b_1 - b_2) x + (c_1 - c_2) &= 0 \end{aligned} $$ ## Analyze the number of real solutions $$ \begin{aligned} A &= a_1 - a_2 > 0 \quad (\text{since } a_1 > 0 \text{ and } a_2 < 0) \ D &= B^2 - 4AC \ D &> 0 \implies 2 \text{ intersections} \ D &= 0 \implies 1 \text{ intersection} \ D &< 0 \implies 0 \text{ intersections} \end{aligned} $$

(A) 0

(B) 1

(C) 2

(D) infinite

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QUESTION IMAGE

Question

Explanation:

Formulate the system of quadratic equations

Set up the intersection equation

Analyze the number of real solutions

Answer: