Question

fully factorise \\(x^2 + 7x + 6\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Quadratic Factorisation",
"Trinomial Factoring"
],
"current_concepts": [
"Quadratic Factorisation",
"Trinomial Factoring"
]
}
</pre_analysis>

<reasoning>

Identify coefficients of the quadratic expression

\[

$$\begin{aligned} &a = 1 \\ &b = 7 \\ &c = 6 \end{aligned}$$

\]

Find two numbers that multiply to \(ac\) and add to \(b\)

\[

$$\begin{aligned} &p \cdot q = 1 \cdot 6 = 6 \\ &p + q = 7 \\ &p = 6, \quad q = 1 \end{aligned}$$

\]

Write the expression in factored form

\[

$$\begin{aligned} &x^2 + 7x + 6 = (x + 6)(x + 1) \end{aligned}$$

\]
</reasoning>

<answer>
\((x + 6)(x + 1)\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Factorisation"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Quadratic Factorisation",
"Trinomial Factoring"
],
"current_concepts": [
"Quadratic Factorisation",
"Trinomial Factoring"
]
}
</pre_analysis>

<reasoning>

Identify coefficients of the quadratic expression

\[

$$\begin{aligned} &a = 1 \\ &b = 7 \\ &c = 6 \end{aligned}$$

\]

Find two numbers that multiply to \(ac\) and add to \(b\)

\[

$$\begin{aligned} &p \cdot q = 1 \cdot 6 = 6 \\ &p + q = 7 \\ &p = 6, \quad q = 1 \end{aligned}$$

\]

Write the expression in factored form

\[

$$\begin{aligned} &x^2 + 7x + 6 = (x + 6)(x + 1) \end{aligned}$$

\]
</reasoning>

<answer>
\((x + 6)(x + 1)\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Factorisation"
]
}
</post_analysis>