Question

identifying the factored form of a trinomial

which expression is the factored form of (x^2 - 7x + 10)?

(x + 3)(x + 4)
(x - 3)(x - 4)
(x - 2)(x - 5)
(x + 2)(x + 5)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Factoring Trinomials",
"Algebra Tiles"
],
"new_concepts": [],
"current_concepts": [
"Factoring Trinomials",
"Algebra Tiles"
]
}
</pre_analysis>

<reasoning>

Identify the trinomial to factor

The given trinomial is:
\[
x^2 - 7x + 10
\]

Find two numbers that multiply to 10 and add to -7

We need factors \(a\) and \(b\) such that:
\[
a \cdot b = 10 \quad \text{and} \quad a + b = -7
\]
The factors of \(10\) that add up to \(-7\) are \(-2\) and \(-5\):
\[
(-2) \cdot (-5) = 10
\]
\[
(-2) + (-5) = -7
\]

Write the factored form

Using the factors \(-2\) and \(-5\), we write the trinomial in its factored form:
\[
x^2 - 7x + 10 = (x - 2)(x - 5)
\]
This matches the algebra tile model shown, which has dimensions of \((x - 2)\) and \((x - 5)\).
</reasoning>

<answer>
<mcq-option>(A) \((x + 3)(x + 4)\)</mcq-option>
<mcq-option>(B) \((x - 3)(x - 4)\)</mcq-option>
<mcq-correct>(C) \((x - 2)(x - 5)\)</mcq-correct>
<mcq-option>(D) \((x + 2)(x + 5)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Factoring Trinomials"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Factoring Trinomials",
"Algebra Tiles"
],
"new_concepts": [],
"current_concepts": [
"Factoring Trinomials",
"Algebra Tiles"
]
}
</pre_analysis>

<reasoning>

Identify the trinomial to factor

The given trinomial is:
\[
x^2 - 7x + 10
\]

Find two numbers that multiply to 10 and add to -7

We need factors \(a\) and \(b\) such that:
\[
a \cdot b = 10 \quad \text{and} \quad a + b = -7
\]
The factors of \(10\) that add up to \(-7\) are \(-2\) and \(-5\):
\[
(-2) \cdot (-5) = 10
\]
\[
(-2) + (-5) = -7
\]

Write the factored form

Using the factors \(-2\) and \(-5\), we write the trinomial in its factored form:
\[
x^2 - 7x + 10 = (x - 2)(x - 5)
\]
This matches the algebra tile model shown, which has dimensions of \((x - 2)\) and \((x - 5)\).
</reasoning>

<answer>
<mcq-option>(A) \((x + 3)(x + 4)\)</mcq-option>
<mcq-option>(B) \((x - 3)(x - 4)\)</mcq-option>
<mcq-correct>(C) \((x - 2)(x - 5)\)</mcq-correct>
<mcq-option>(D) \((x + 2)(x + 5)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Factoring Trinomials"
]
}
</post_analysis>