what are the solutions of $\frac{1}{4}x^2 = -\frac{1}{2}x + 2$?

-4 and 2
-4 and 1
0 and 4
1 and 4

Question

what are the solutions of $\frac{1}{4}x^2 = -\frac{1}{2}x + 2$?

-4 and 2
-4 and 1
0 and 4
1 and 4

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", "Graphical Solution of Equations" ], "new_concepts": [], "current_concepts": [ "Solving Quadratic Equations", "Graphical Solution of Equations" ] } ## Identify intersection points from the graph $$ \begin{aligned} &y_1 = \frac{1}{4}x^2 \ &y_2 = -\frac{1}{2}x + 2 \ &\text{Intersection points: } (-4, 4) \text{ and } (2, 1) \ &x\text{-coordinates of intersections: } x = -4, \quad x = 2 \end{aligned} $$ ## Solve the quadratic equation algebraically $$ \begin{aligned} &\frac{1}{4}x^2 = -\frac{1}{2}x + 2 \ &x^2 = -2x + 8 \ &x^2 + 2x - 8 = 0 \ &(x + 4)(x - 2) = 0 \ &x = -4 \quad \text{or} \quad x = 2 \end{aligned} $$

-4 and 2

-4 and 1

0 and 4

1 and 4

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

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QUESTION IMAGE

Question

Explanation:

Identify intersection points from the graph

Solve the quadratic equation algebraically

Answer: