Question

which shows the factored form of (x^2 - 12x - 45)?

(x + 3)(x - 15)
(x - 3)(x - 15)
(x + 3)(x + 15)
(x - 3)(x + 15)

Explanation:

<pre_analysis>
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<reasoning>

Identify the coefficients of the quadratic expression

\[

$$\begin{aligned} &a = 1\\ &b = -12\\ &c = -45 \end{aligned}$$

\]

Find two numbers that multiply to c and add to b

\[

$$\begin{aligned} &p \cdot q = -45\\ &p + q = -12\\ &\text{Factors of } -45: \quad 3 \text{ and } -15\\ &3 \cdot (-15) = -45\\ &3 + (-15) = -12 \end{aligned}$$

\]

Write the expression in factored form

\[
x^2 - 12x - 45 = (x + 3)(x - 15)
\]
</reasoning>

<answer>
<mcq-correct>(x + 3)(x - 15)</mcq-correct>
<mcq-option>(x - 3)(x - 15)</mcq-option>
<mcq-option>(x + 3)(x + 15)</mcq-option>
<mcq-option>(x - 3)(x + 15)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Factoring Trinomials"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Identify the coefficients of the quadratic expression

\[

$$\begin{aligned} &a = 1\\ &b = -12\\ &c = -45 \end{aligned}$$

\]

Find two numbers that multiply to c and add to b

\[

$$\begin{aligned} &p \cdot q = -45\\ &p + q = -12\\ &\text{Factors of } -45: \quad 3 \text{ and } -15\\ &3 \cdot (-15) = -45\\ &3 + (-15) = -12 \end{aligned}$$

\]

Write the expression in factored form

\[
x^2 - 12x - 45 = (x + 3)(x - 15)
\]
</reasoning>

<answer>
<mcq-correct>(x + 3)(x - 15)</mcq-correct>
<mcq-option>(x - 3)(x - 15)</mcq-option>
<mcq-option>(x + 3)(x + 15)</mcq-option>
<mcq-option>(x - 3)(x + 15)</mcq-option>
</answer>

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"Algebra",
"Factoring Trinomials"
]
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