Question

if the roots of the quadratic equation are complex, then ______.

\\(\delta < 0\\)

\\(\delta = 0\\)

Explanation:

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<reasoning>

Define the discriminant of a quadratic equation

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is denoted by \(\Delta\) (or \(D\)) and is given by:
\[
\Delta = b^2 - 4ac
\]

Determine the condition for complex roots

The nature of the roots depends on the sign of the discriminant \(\Delta\):

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), the equation has one repeated real root.
• If \(\Delta < 0\), the equation has two complex (conjugate) roots.

</reasoning>

<answer>
<mcq-correct>\(\Delta < 0\)</mcq-correct>
<mcq-option>\(\Delta = 0\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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"Mathematics",
"Algebra",
"Discriminant of Quadratic"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Define the discriminant of a quadratic equation

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is denoted by \(\Delta\) (or \(D\)) and is given by:
\[
\Delta = b^2 - 4ac
\]

Determine the condition for complex roots

The nature of the roots depends on the sign of the discriminant \(\Delta\):

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), the equation has one repeated real root.
• If \(\Delta < 0\), the equation has two complex (conjugate) roots.

</reasoning>

<answer>
<mcq-correct>\(\Delta < 0\)</mcq-correct>
<mcq-option>\(\Delta = 0\)</mcq-option>
</answer>

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"Mathematics",
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