how many real zeros does the following quadratic function have?

\f(x) = -4x^2 + 12x - 9\

remember the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

* one real zero
* no real zeros
* two real zeros

Question

how many real zeros does the following quadratic function have?

\f(x) = -4x^2 + 12x - 9\

remember the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

* one real zero
* no real zeros
* two real zeros

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Discriminant Analysis", "Number of Quadratic Solutions" ], "new_concepts": [], "current_concepts": [ "Discriminant Analysis", "Number of Quadratic Solutions", "Quadratic Zeros" ] } ## Identify coefficients of the quadratic function Using the **Discriminant Analysis** and **Number of Quadratic Solutions** knowledge points $$ \begin{aligned} &f(x) = -4x^2 + 12x - 9\ &a = -4,\quad b = 12,\quad c = -9 \end{aligned} $$ ## Calculate the discriminant Using the **Discriminant Analysis** knowledge point $$ \begin{aligned} D &= b^2 - 4ac\ D &= 12^2 - 4(-4)(-9)\ D &= 144 - 144\ D &= 0 \end{aligned} $$ ## Determine the number of real zeros Using the **Number of Quadratic Solutions** knowledge point $$ \begin{aligned} &D = 0 \implies \text{Exactly one real zero} \end{aligned} $$

one real zero

no real zeros

two real zeros

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Discriminant Analysis" ] }

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Question

Explanation:

Identify coefficients of the quadratic function

Calculate the discriminant

Determine the number of real zeros

Answer: