Question

use factoring to solve the quadratic equation. check by substitution or by using a graphing utility and identifying x-intercepts.

\\2x^2 + 12x = 0\\

rewrite the equation in factored form.

\\\box = 0\\
(factor completely.)

the solution set is \\(\\{\box\\}\\).
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each solution only once.)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quadratic Equations",
"Zero-Product Principle"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Equations",
"Factoring Quadratic Trinomials",
"Zero-Product Principle"
]
}
</pre_analysis>

<reasoning>

Identify the common factors

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &\text{Given equation: } 2x^2 + 12x = 0\\ &\text{Greatest Common Factor (GCF) of } 2x^2 \text{ and } 12x \text{ is } 2x \end{aligned}$$

\]

Factor the quadratic expression

Using the Quadratic Equations knowledge point
\[
2x(x + 6) = 0
\]

Apply the zero-product principle

Using the Zero-Product Principle knowledge point
\[

$$\begin{aligned} &2x = 0 \quad \text{or} \quad x + 6 = 0\\ &x = 0 \quad \text{or} \quad x = -6 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

Rewrite the equation in factored form.
<blank>\(2x(x + 6)\)</blank> \( = 0\)

Question 2

The solution set is {<blank>\(0, -6\)</blank>}.
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quadratic Equations",
"Zero-Product Principle"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Equations",
"Factoring Quadratic Trinomials",
"Zero-Product Principle"
]
}
</pre_analysis>

<reasoning>

Identify the common factors

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &\text{Given equation: } 2x^2 + 12x = 0\\ &\text{Greatest Common Factor (GCF) of } 2x^2 \text{ and } 12x \text{ is } 2x \end{aligned}$$

\]

Factor the quadratic expression

Using the Quadratic Equations knowledge point
\[
2x(x + 6) = 0
\]

Apply the zero-product principle

Using the Zero-Product Principle knowledge point
\[

$$\begin{aligned} &2x = 0 \quad \text{or} \quad x + 6 = 0\\ &x = 0 \quad \text{or} \quad x = -6 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

Rewrite the equation in factored form.
<blank>\(2x(x + 6)\)</blank> \( = 0\)

Question 2

The solution set is {<blank>\(0, -6\)</blank>}.
</answer>

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