which of the statements about the following quadratic equation is true?
\6x^2 - 8 = 4x^2 + 7x\
- the discriminant is greater than zero, so there are two real roots.
- the discriminant is greater than zero, so there are two complex roots.
- the discriminant is less than zero, so there are two real roots.
- the discriminant is less than zero, so there are two complex roots.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Solving Quadratic Equations" ], "new_concepts": [ "Discriminant of Quadratic Equation", "Nature of Quadratic Roots" ], "current_concepts": [ "Solving Quadratic Equations", "Discriminant of Quadratic Equation", "Nature of Quadratic Roots" ] } ## Write the quadratic equation in standard form $$ \begin{aligned} 6x^2 - 8 &= 4x^2 + 7x \ 6x^2 - 4x^2 - 7x - 8 &= 0 \ 2x^2 - 7x - 8 &= 0 \end{aligned} $$ ## Calculate the discriminant $$ \begin{aligned} a &= 2, \quad b = -7, \quad c = -8 \ D &= b^2 - 4ac \ D &= (-7)^2 - 4(2)(-8) \ D &= 49 + 64 \ D &= 113 \end{aligned} $$ ## Determine the nature of the roots $$ \begin{aligned} D &= 113 > 0 \implies \text{Two real roots} \end{aligned} $$

(A) The discriminant is greater than zero, so there are two real roots.

(B) The discriminant is greater than zero, so there are two complex roots.

(C) The discriminant is less than zero, so there are two real roots.

(D) The discriminant is less than zero, so there are two complex roots.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Discriminant of Quadratic Equation" ] }

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Explanation:

Write the quadratic equation in standard form

Calculate the discriminant

Determine the nature of the roots

Answer: