Question

fill in the blank so that the resulting statement is true.

the solutions of a quadratic equation in standard form \\(ax^2 + bx + c = 0\\), \\(a \
eq 0\\), are given by the quadratic formula \\(x = \underline{\quad\quad\quad}\\)

the solutions of a quadratic equation in standard form \\(ax^2 + bx + c = 0\\), \\(a \
eq 0\\), are given by the quadratic formula \\(x = \\)

options:

1. \\(\frac{-b - \sqrt{b^2 - ac}}{2a}\\)
2. \\(\frac{-b + \sqrt{b^2 - 4ac}}{a}\\)
3. \\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\)
4. \\(\frac{-b \pm \sqrt{b^2 - ac}}{2a}\\)

Explanation:

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"concepts_used": [
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</pre_analysis>

<reasoning>

Identify the standard form

Using the Quadratic Equations knowledge point
\[
ax^2 + bx + c = 0, \quad a
eq 0
\]

State the quadratic formula

The standard formula derived by completing the square is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Match with the given options

Comparing the standard formula to the dropdown choices:

• Option 1: \(\frac{-b - \sqrt{b^2 - ac}}{2a}\) (Incorrect)
• Option 2: \(\frac{-b + \sqrt{b^2 - 4ac}}{a}\) (Incorrect)
• Option 3: \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) (Correct)
• Option 4: \(\frac{-b \pm \sqrt{b^2 - ac}}{2a}\) (Incorrect)

</reasoning>

<answer>
The solutions of a quadratic equation in standard form \(ax^2 + bx + c = 0\), \(a
eq 0\), are given by the quadratic formula \(x =\) <blank>\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Formula"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
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],
"needs_drawing": false,
"concepts_used": [
"Quadratic Equations"
],
"new_concepts": [
"Quadratic Formula"
],
"current_concepts": [
"Quadratic Equations",
"Quadratic Formula"
]
}
</pre_analysis>

<reasoning>

Identify the standard form

Using the Quadratic Equations knowledge point
\[
ax^2 + bx + c = 0, \quad a
eq 0
\]

State the quadratic formula

The standard formula derived by completing the square is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Match with the given options

Comparing the standard formula to the dropdown choices:

• Option 1: \(\frac{-b - \sqrt{b^2 - ac}}{2a}\) (Incorrect)
• Option 2: \(\frac{-b + \sqrt{b^2 - 4ac}}{a}\) (Incorrect)
• Option 3: \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) (Correct)
• Option 4: \(\frac{-b \pm \sqrt{b^2 - ac}}{2a}\) (Incorrect)

</reasoning>

<answer>
The solutions of a quadratic equation in standard form \(ax^2 + bx + c = 0\), \(a
eq 0\), are given by the quadratic formula \(x =\) <blank>\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Formula"
]
}
</post_analysis>