Question

value: 1

the product of two consecutive positive integers is 72. find the two positive integers.

a. 8 and -9
b. -8 and -9
c. -8 and 9
d. 8 and 9

Explanation:

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<reasoning>

Set up the quadratic equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &x(x + 1) = 72 \\ &x^2 + x - 72 = 0 \end{aligned}$$

\]

Solve the quadratic equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &(x + 9)(x - 8) = 0 \\ &x = -9 \quad \text{or} \quad x = 8 \end{aligned}$$

\]

Determine the positive integer solutions

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &\text{Since } x \text{ must be a positive integer:} \\ &x = 8 \\ &x + 1 = 9 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a. 8 and -9</mcq-option>
<mcq-option>b. -8 and -9</mcq-option>
<mcq-option>c. -8 and 9</mcq-option>
<mcq-correct>d. 8 and 9</mcq-correct>
</answer>

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

Answer:

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<reasoning>

Set up the quadratic equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &x(x + 1) = 72 \\ &x^2 + x - 72 = 0 \end{aligned}$$

\]

Solve the quadratic equation

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &(x + 9)(x - 8) = 0 \\ &x = -9 \quad \text{or} \quad x = 8 \end{aligned}$$

\]

Determine the positive integer solutions

Using the Quadratic Equations knowledge point
\[

$$\begin{aligned} &\text{Since } x \text{ must be a positive integer:} \\ &x = 8 \\ &x + 1 = 9 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>a. 8 and -9</mcq-option>
<mcq-option>b. -8 and -9</mcq-option>
<mcq-option>c. -8 and 9</mcq-option>
<mcq-correct>d. 8 and 9</mcq-correct>
</answer>

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