Question

simplify the following expression.

\\(5x - 2)^2\\

\\?x^2 + \quadx + \quad\\

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": [
"Multiplying Binomials",
"Combining Like Terms"
],
"new_concepts": [
"Perfect Square Trinomials"
],
"current_concepts": [
"Multiplying Binomials",
"Combining Like Terms",
"Perfect Square Trinomials"
]
}
</pre_analysis>

<reasoning>

Write the squared expression as a product

Using the Multiplying Binomials knowledge point
\[
(5x - 2)^2 = (5x - 2)(5x - 2)
\]

Expand the binomial product

Using the Multiplying Binomials knowledge point
\[
(5x - 2)(5x - 2) = 5x(5x) + 5x(-2) - 2(5x) - 2(-2)
\]
\[
= 25x^2 - 10x - 10x + 4
\]

Combine the linear terms

Using the Combining Like Terms knowledge point
\[
25x^2 - 10x - 10x + 4 = 25x^2 - 20x + 4
\]
</reasoning>

<answer>
Simplify the following expression.
\[(5x - 2)^2\]
<blank>25</blank>\(x^2 +\) <blank>-20</blank>\(x +\) <blank>4</blank>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Multiplying Binomials"
]
}
</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": [
"Multiplying Binomials",
"Combining Like Terms"
],
"new_concepts": [
"Perfect Square Trinomials"
],
"current_concepts": [
"Multiplying Binomials",
"Combining Like Terms",
"Perfect Square Trinomials"
]
}
</pre_analysis>

<reasoning>

Write the squared expression as a product

Using the Multiplying Binomials knowledge point
\[
(5x - 2)^2 = (5x - 2)(5x - 2)
\]

Expand the binomial product

Using the Multiplying Binomials knowledge point
\[
(5x - 2)(5x - 2) = 5x(5x) + 5x(-2) - 2(5x) - 2(-2)
\]
\[
= 25x^2 - 10x - 10x + 4
\]

Combine the linear terms

Using the Combining Like Terms knowledge point
\[
25x^2 - 10x - 10x + 4 = 25x^2 - 20x + 4
\]
</reasoning>

<answer>
Simplify the following expression.
\[(5x - 2)^2\]
<blank>25</blank>\(x^2 +\) <blank>-20</blank>\(x +\) <blank>4</blank>
</answer>

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