determine the number of real solutions each quadratic equation has

$y = 12x^2 - 9x + 4$
real solution(s)

$4y - 7 = 5x^2 - x + 2 + 3y$
real solution(s)

$10x + y = -x^2 + 2$
real solution(s)

$y = (-x + 4)^2$
real solution(s)

no
one
two

Question

determine the number of real solutions each quadratic equation has

$y = 12x^2 - 9x + 4$
real solution(s)

$4y - 7 = 5x^2 - x + 2 + 3y$
real solution(s)

$10x + y = -x^2 + 2$
real solution(s)

$y = (-x + 4)^2$
real solution(s)

no
one
two

Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Discriminant of Quadratic Equation", "Nature of Quadratic Roots" ], "new_concepts": [], "current_concepts": [ "Discriminant of Quadratic Equation", "Nature of Quadratic Roots" ] } ## Analyze the equation $y = 12x^2 - 9x + 4$ when $y = 0$ $$ \begin{aligned} &12x^2 - 9x + 4 = 0 \ &D = (-9)^2 - 4(12)(4) = 81 - 192 = -111 < 0 \end{aligned} $$ ## Analyze the equation $4y - 7 = 5x^2 - x + 2 + 3y$ when $y = 0$ $$ \begin{aligned} &4(0) - 7 = 5x^2 - x + 2 + 3(0) \ &-7 = 5x^2 - x + 2 \ &5x^2 - x + 9 = 0 \ &D = (-1)^2 - 4(5)(9) = 1 - 180 = -179 < 0 \end{aligned} $$ ## Analyze the equations $10x + y = -x^2 + 2$ and $y = (-x + 4)^2$ when $y = 0$ $$ \begin{aligned} &\text{For } 10x + 0 = -x^2 + 2: \ &x^2 + 10x - 2 = 0 \ &D = 10^2 - 4(1)(-2) = 100 + 8 = 108 > 0 \ \ &\text{For } 0 = (-x + 4)^2: \ &(x - 4)^2 = 0 \ &x = 4 \quad (\text{double root}) \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 1 | $y = 12x^2 - 9x + 4$ | no | | 2 | $4y - 7 = 5x^2 - x + 2 + 3y$ | no | | 3 | $10x + y = -x^2 + 2$ | two | | 4 | $y = (-x + 4)^2$ | one | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Discriminant of Quadratic Equation" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze the equation \(y = 12x^2 - 9x + 4\) when \(y = 0\)

Analyze the equation \(4y - 7 = 5x^2 - x + 2 + 3y\) when \(y = 0\)

Analyze the equations \(10x + y = -x^2 + 2\) and \(y = (-x + 4)^2\) when \(y = 0\)

Answer:

No.	Problem	Answer
2	\(4y - 7 = 5x^2 - x + 2 + 3y\)	no
3	\(10x + y = -x^2 + 2\)	two
4	\(y = (-x + 4)^2\)	one