Question

solve for all possible values of x.

\\(\sqrt{x - 9} + 4 = 8\\)

• \\(x = -5\\)
• \\(x = 7\\)
• \\(x = 13\\)
• \\(x = 25\\)

question 2 (multiple choice worth 6 points)
(01.08r mc)

simplify \\((8j^3 + 3j^2 + 7) - (3j^3 - 7j^2 - 7j - 12)\\).

• \\(5j^3 - 4j^2 - 7j - 5\\)
• \\(5j^3 + 10j^2 - 7j - 5\\)
• \\(5j^3 - 4j^2 + 7j + 19\\)
• \\(5j^3 + 10j^2 + 7j + 19\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Radical Equations"
],
"new_concepts": [
"Polynomial Subtraction"
],
"current_concepts": [
"Radical Equations",
"Polynomial Subtraction"
]
}
</pre_analysis>

<reasoning>

Isolate the radical term

Using the Radical Equations knowledge point
\[

$$\begin{aligned} \sqrt{x-9} + 4 &= 8 \\ \sqrt{x-9} &= 4 \end{aligned}$$

\]

Square both sides to solve

Using the Radical Equations knowledge point
\[

$$\begin{aligned} (\sqrt{x-9})^2 &= 4^2 \\ x - 9 &= 16 \\ x &= 25 \end{aligned}$$

\]

Distribute the negative sign

Distribute the negative sign to each term in the second polynomial.
\[
-(3j^3 - 7j^2 - 7j - 12) = -3j^3 + 7j^2 + 7j + 12
\]

Combine the like terms

Group and combine the coefficients of the terms with matching powers.
\[

$$\begin{aligned} (8j^3 - 3j^3) + (3j^2 + 7j^2) + 7j + (7 + 12) \\ 5j^3 + 10j^2 + 7j + 19 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(x = -5\)</mcq-option>
<mcq-option>(B) \(x = 7\)</mcq-option>
<mcq-option>(C) \(x = 13\)</mcq-option>
<mcq-correct>(D) \(x = 25\)</mcq-correct>

Question 2

<mcq-option>(A) \(5j^3 - 4j^2 - 7j - 5\)</mcq-option>
<mcq-option>(B) \(5j^3 + 10j^2 - 7j - 5\)</mcq-option>
<mcq-option>(C) \(5j^3 - 4j^2 + 7j + 19\)</mcq-option>
<mcq-correct>(D) \(5j^3 + 10j^2 + 7j + 19\)</mcq-correct>
</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Radical Equations"
],
"new_concepts": [
"Polynomial Subtraction"
],
"current_concepts": [
"Radical Equations",
"Polynomial Subtraction"
]
}
</pre_analysis>

<reasoning>

Isolate the radical term

Using the Radical Equations knowledge point
\[

$$\begin{aligned} \sqrt{x-9} + 4 &= 8 \\ \sqrt{x-9} &= 4 \end{aligned}$$

\]

Square both sides to solve

Using the Radical Equations knowledge point
\[

$$\begin{aligned} (\sqrt{x-9})^2 &= 4^2 \\ x - 9 &= 16 \\ x &= 25 \end{aligned}$$

\]

Distribute the negative sign

Distribute the negative sign to each term in the second polynomial.
\[
-(3j^3 - 7j^2 - 7j - 12) = -3j^3 + 7j^2 + 7j + 12
\]

Combine the like terms

Group and combine the coefficients of the terms with matching powers.
\[

$$\begin{aligned} (8j^3 - 3j^3) + (3j^2 + 7j^2) + 7j + (7 + 12) \\ 5j^3 + 10j^2 + 7j + 19 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(x = -5\)</mcq-option>
<mcq-option>(B) \(x = 7\)</mcq-option>
<mcq-option>(C) \(x = 13\)</mcq-option>
<mcq-correct>(D) \(x = 25\)</mcq-correct>

Question 2

<mcq-option>(A) \(5j^3 - 4j^2 - 7j - 5\)</mcq-option>
<mcq-option>(B) \(5j^3 + 10j^2 - 7j - 5\)</mcq-option>
<mcq-option>(C) \(5j^3 - 4j^2 + 7j + 19\)</mcq-option>
<mcq-correct>(D) \(5j^3 + 10j^2 + 7j + 19\)</mcq-correct>
</answer>

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