Question

question

solve for \\(x\\) by factoring: \\(5x^2 + 36x + 7 = 0\\).

write the two solutions in the answer template below.

provide your answer below:

\\(x = \square, \square\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"AC Method",
"Factoring by Grouping",
"Zero Product Property",
"Solving Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"AC Method",
"Factoring by Grouping",
"Zero Product Property",
"Solving Linear Equations"
]
}
</pre_analysis>

<reasoning>

Identify the coefficients

Using the AC Method knowledge point
\[

$$\begin{aligned} &a = 5,\quad b = 36,\quad c = 7\\ &a \cdot c = 5 \cdot 7 = 35 \end{aligned}$$

\]

Split the middle term

Using the AC Method knowledge point
\[

$$\begin{aligned} &35 \cdot 1 = 35\\ &35 + 1 = 36\\ &5x^2 + 35x + x + 7 = 0 \end{aligned}$$

\]

Factor by grouping

Using the Factoring by Grouping knowledge point
\[

$$\begin{aligned} &5x(x + 7) + 1(x + 7) = 0\\ &(5x + 1)(x + 7) = 0 \end{aligned}$$

\]

Apply the zero product property

Using the Zero Product Property knowledge point
\[

$$\begin{aligned} &5x + 1 = 0 \quad \text{or} \quad x + 7 = 0 \end{aligned}$$

\]

Solve for x

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} &x = -\frac{1}{5},\quad x = -7 \end{aligned}$$

\]
</reasoning>

<answer>
Solve for \(x\) by factoring: \(5x^2 + 36x + 7 = 0\).

Write the two solutions in the answer template below.

Provide your answer below:

\(x =\) <blank>\(-\frac{1}{5}\)</blank>, <blank>\(-7\)</blank>
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"AC Method",
"Factoring by Grouping",
"Zero Product Property",
"Solving Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"AC Method",
"Factoring by Grouping",
"Zero Product Property",
"Solving Linear Equations"
]
}
</pre_analysis>

<reasoning>

Identify the coefficients

Using the AC Method knowledge point
\[

$$\begin{aligned} &a = 5,\quad b = 36,\quad c = 7\\ &a \cdot c = 5 \cdot 7 = 35 \end{aligned}$$

\]

Split the middle term

Using the AC Method knowledge point
\[

$$\begin{aligned} &35 \cdot 1 = 35\\ &35 + 1 = 36\\ &5x^2 + 35x + x + 7 = 0 \end{aligned}$$

\]

Factor by grouping

Using the Factoring by Grouping knowledge point
\[

$$\begin{aligned} &5x(x + 7) + 1(x + 7) = 0\\ &(5x + 1)(x + 7) = 0 \end{aligned}$$

\]

Apply the zero product property

Using the Zero Product Property knowledge point
\[

$$\begin{aligned} &5x + 1 = 0 \quad \text{or} \quad x + 7 = 0 \end{aligned}$$

\]

Solve for x

Using the Solving Linear Equations knowledge point
\[

$$\begin{aligned} &x = -\frac{1}{5},\quad x = -7 \end{aligned}$$

\]
</reasoning>

<answer>
Solve for \(x\) by factoring: \(5x^2 + 36x + 7 = 0\).

Write the two solutions in the answer template below.

Provide your answer below:

\(x =\) <blank>\(-\frac{1}{5}\)</blank>, <blank>\(-7\)</blank>
</answer>

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