how many real zeros does the following quadratic function have?

\f(x) = -x^2 - 6x - 7\

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

no real zeros
one real zero
two real zeros

Question

how many real zeros does the following quadratic function have?

\f(x) = -x^2 - 6x - 7\

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

no real zeros
one real zero
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": [ "Standard Form of a Quadratic", "Number of Quadratic Solutions" ], "new_concepts": [ "Discriminant Analysis" ], "current_concepts": [ "Standard Form of a Quadratic", "Number of Quadratic Solutions", "Discriminant Analysis" ] } ## Identify coefficients in standard form Using the *Standard Form of a Quadratic* knowledge point $$ \begin{aligned} &f(x) = ax^2 + bx + c \ &a = -1, \quad b = -6, \quad c = -7 \end{aligned} $$ ## Calculate the discriminant Using the *Number of Quadratic Solutions* knowledge point $$ \begin{aligned} &D = b^2 - 4ac \ &D = (-6)^2 - 4(-1)(-7) \ &D = 36 - 28 \ &D = 8 \end{aligned} $$ ## Determine the number of real zeros Since the discriminant $D = 8$ is strictly greater than zero ($D > 0$), the quadratic equation has exactly two distinct real solutions. Therefore, the function has two real zeros.

no real zeros

one real zero

two real zeros

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

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Question

Explanation:

Identify coefficients in standard form

Calculate the discriminant

Determine the number of real zeros

Answer: