drag each equation to the correct location on the table.

for each equation, determine the number of solutions and place on the appropriate field in the table.

table columns:
- no real solutions
- exactly one real solution
- two real solutions

equations to drag:
(5x^2 + 2 = 4x)
(3(x + 5)^2 = -2)
(4x^2 - 16x = 0)
(3x^2 + 24x = -48)

Question

drag each equation to the correct location on the table.

for each equation, determine the number of solutions and place on the appropriate field in the table.

table columns:
- no real solutions
- exactly one real solution
- two real solutions

equations to drag:
(5x^2 + 2 = 4x)
(3(x + 5)^2 = -2)
(4x^2 - 16x = 0)
(3x^2 + 24x = -48)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Solving Quadratic Equations", "Discriminant", "Complex Roots" ], "new_concepts": [], "current_concepts": [ "Solving Quadratic Equations", "Discriminant", "Complex Roots" ] } ## Analyze the discriminant of the first and third equations $$ \begin{aligned} &5x^2 - 4x + 2 = 0 \implies D = (-4)^2 - 4(5)(2) = 16 - 40 = -24 < 0 \quad \text{(No Real Solutions)} \ &4x^2 - 16x = 0 \implies 4x(x - 4) = 0 \implies x = 0, 4 \quad \text{(Two Real Solutions)} \end{aligned} $$ ## Analyze the second and fourth equations $$ \begin{aligned} &3(x + 5)^2 = -2 \implies (x + 5)^2 = -\frac{2}{3} < 0 \quad \text{(No Real Solutions)} \ &3x^2 + 24x + 48 = 0 \implies 3(x^2 + 8x + 16) = 0 \implies 3(x + 4)^2 = 0 \implies x = -4 \quad \text{(Exactly One Real Solution)} \end{aligned} $$ | No Real Solutions | Exactly One Real Solution | Two Real Solutions | |---|---|---| | $5x^2 + 2 = 4x$ | $3x^2 + 24x = -48$ | $4x^2 - 16x = 0$ | | $3(x + 5)^2 = -2$ | | | { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Discriminant" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze the discriminant of the first and third equations

Analyze the second and fourth equations

Answer: