Question

which function has zeros at \\(x = -2\\) and \\(x = 5\\)?

\\(f(x) = x^2 - 3x - 10\\)

\\(f(x) = x^2 + 3x - 10\\)

\\(f(x) = x^2 + 2x - 10\\)

Explanation:

<pre_analysis>
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<reasoning>

Identify the given zeros

\[
x_1 = -2,\quad x_2 = 5
\]

Write the factored form of the quadratic function

\[
f(x) = (x - x_1)(x - x_2) = (x - (-2))(x - 5) = (x + 2)(x - 5)
\]

Expand the factored expression

\[
f(x) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10
\]
</reasoning>

<answer>
<mcq-correct>(A) \(f(x) = x^2 - 3x - 10\)</mcq-correct>
<mcq-option>(B) \(f(x) = x^2 + 3x - 10\)</mcq-option>
<mcq-option>(C) \(f(x) = x^2 + 2x - 10\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Roots of Quadratic Functions"
]
}
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Answer:

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</pre_analysis>

<reasoning>

Identify the given zeros

\[
x_1 = -2,\quad x_2 = 5
\]

Write the factored form of the quadratic function

\[
f(x) = (x - x_1)(x - x_2) = (x - (-2))(x - 5) = (x + 2)(x - 5)
\]

Expand the factored expression

\[
f(x) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10
\]
</reasoning>

<answer>
<mcq-correct>(A) \(f(x) = x^2 - 3x - 10\)</mcq-correct>
<mcq-option>(B) \(f(x) = x^2 + 3x - 10\)</mcq-option>
<mcq-option>(C) \(f(x) = x^2 + 2x - 10\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Roots of Quadratic Functions"
]
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