Question

solving quadratic equations: zero product property

a quadratic equation has the solutions \\(x = -4\\) and \\(x = -7\\).

write a possible quadratic equation in standard form.

enter your answers in the boxes.

\\(x^2 + \square x + \square = 0\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Zero Product Property",
"Roots of Quadratic Functions"
],
"new_concepts": [],
"current_concepts": [
"Zero Product Property",
"Roots of Quadratic Functions",
"Factoring Quadratics"
]
}
</pre_analysis>

<reasoning>

Write the factored form of the quadratic equation

\[
(x - (-4))(x - (-7)) = 0
\]
\[
(x + 4)(x + 7) = 0
\]

Expand the factors to standard form

\[
x^2 + 7x + 4x + 28 = 0
\]
\[
x^2 + 11x + 28 = 0
\]

Identify the coefficients for the boxes

\[
x^2 + 11x + 28 = 0
\]
</reasoning>

<answer>
A quadratic equation has the solutions \(x = -4\) and \(x = -7\).

Write a possible quadratic equation in standard form.

Enter your answers in the boxes.

\(x^2 +\) <blank>11</blank> \(x +\) <blank>28</blank> \(= 0\)
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Zero Product Property",
"Roots of Quadratic Functions"
],
"new_concepts": [],
"current_concepts": [
"Zero Product Property",
"Roots of Quadratic Functions",
"Factoring Quadratics"
]
}
</pre_analysis>

<reasoning>

Write the factored form of the quadratic equation

\[
(x - (-4))(x - (-7)) = 0
\]
\[
(x + 4)(x + 7) = 0
\]

Expand the factors to standard form

\[
x^2 + 7x + 4x + 28 = 0
\]
\[
x^2 + 11x + 28 = 0
\]

Identify the coefficients for the boxes

\[
x^2 + 11x + 28 = 0
\]
</reasoning>

<answer>
A quadratic equation has the solutions \(x = -4\) and \(x = -7\).

Write a possible quadratic equation in standard form.

Enter your answers in the boxes.

\(x^2 +\) <blank>11</blank> \(x +\) <blank>28</blank> \(= 0\)
</answer>

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