Question

drag and drop the following choices into the boxes given below.

quadratic equation
\\(x^2 = 16\\)
\\(x^2 - 25 = 0\\)
\\(5x^2 - 245 = 0\\)

choices:

• \\(4\\) or \\(-4\\)
• \\(5\\) or \\(-5\\)
• \\(7\\) or \\(-7\\)
• a polynomial equation in which the highest power of the variable is two.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quadratic Equations",
"Difference of Squares"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Equations",
"Difference of Squares"
]
}
</pre_analysis>

<reasoning>

Define quadratic equation

Using the Quadratic Equations knowledge point
\[
ax^2 + bx + c = 0 \quad (a
eq 0)
\]
A polynomial equation in which the highest power of the variable is two.

Solve first equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} x^2 &= 16 \\ x &= \pm 4 \end{aligned}$$

\]

Solve second equation

Using the Difference of Squares knowledge point
\[

$$\begin{aligned} x^2 - 25 &= 0 \\ (x - 5)(x + 5) &= 0 \\ x &= \pm 5 \end{aligned}$$

\]

Solve third equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} 5x^2 - 245 &= 0 \\ 5(x^2 - 49) &= 0 \\ x^2 &= 49 \\ x &= \pm 7 \end{aligned}$$

\]
</reasoning>

<answer>

Left Item	Matched Choice
\(x^2 = 16\)	\(4\) or \(-4\)
\(x^2 - 25 = 0\)	\(5\) or \(-5\)
\(5x^2 - 245 = 0\)	\(7\) or \(-7\)

</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": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quadratic Equations",
"Difference of Squares"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Equations",
"Difference of Squares"
]
}
</pre_analysis>

<reasoning>

Define quadratic equation

Using the Quadratic Equations knowledge point
\[
ax^2 + bx + c = 0 \quad (a
eq 0)
\]
A polynomial equation in which the highest power of the variable is two.

Solve first equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} x^2 &= 16 \\ x &= \pm 4 \end{aligned}$$

\]

Solve second equation

Using the Difference of Squares knowledge point
\[

$$\begin{aligned} x^2 - 25 &= 0 \\ (x - 5)(x + 5) &= 0 \\ x &= \pm 5 \end{aligned}$$

\]

Solve third equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} 5x^2 - 245 &= 0 \\ 5(x^2 - 49) &= 0 \\ x^2 &= 49 \\ x &= \pm 7 \end{aligned}$$

\]
</reasoning>

<answer>

Left Item	Matched Choice
\(x^2 = 16\)	\(4\) or \(-4\)
\(x^2 - 25 = 0\)	\(5\) or \(-5\)
\(5x^2 - 245 = 0\)	\(7\) or \(-7\)

</answer>

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