Question

solving quadratic equation - factorization method

example 4

solve: \\(x^2 - 15x + 56 = 0\\)

solution

since the coefficient of \\(x^2\\) term is 1, the factors can be written as \\((x + a)(x + b) = 0\\).

value: 1

find the two numbers whose sum is -15 and their product is 56.

a. 7 and -8
b. -7 and -8
c. -7 and 8
d. 7 and 8

Explanation:

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

<reasoning>

Identify the sum and product requirements

\[

$$\begin{aligned} &a + b = -15 \\ &a \cdot b = 56 \end{aligned}$$

\]

Test the given options

\[

$$\begin{aligned} &\text{Option a: } 7 + (-8) = -1 eq -15 \\ &\text{Option b: } -7 + (-8) = -15, \quad (-7) \cdot (-8) = 56 \\ &\text{Option c: } -7 + 8 = 1 eq -15 \\ &\text{Option d: } 7 + 8 = 15 eq -15 \end{aligned}$$

\]

Select the correct option

\[
a = -7, \quad b = -8
\]
</reasoning>

<answer>
<mcq-option>a. 7 and -8</mcq-option>
<mcq-correct>b. -7 and -8</mcq-correct>
<mcq-option>c. -7 and 8</mcq-option>
<mcq-option>d. 7 and 8</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Equations"
]
}
</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": [
"Quadratic Equations"
],
"new_concepts": [],
"current_concepts": [
"Quadratic Equations",
"Factoring Quadratics"
]
}
</pre_analysis>

<reasoning>

Identify the sum and product requirements

\[

$$\begin{aligned} &a + b = -15 \\ &a \cdot b = 56 \end{aligned}$$

\]

Test the given options

\[

$$\begin{aligned} &\text{Option a: } 7 + (-8) = -1 eq -15 \\ &\text{Option b: } -7 + (-8) = -15, \quad (-7) \cdot (-8) = 56 \\ &\text{Option c: } -7 + 8 = 1 eq -15 \\ &\text{Option d: } 7 + 8 = 15 eq -15 \end{aligned}$$

\]

Select the correct option

\[
a = -7, \quad b = -8
\]
</reasoning>

<answer>
<mcq-option>a. 7 and -8</mcq-option>
<mcq-correct>b. -7 and -8</mcq-correct>
<mcq-option>c. -7 and 8</mcq-option>
<mcq-option>d. 7 and 8</mcq-option>
</answer>

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