Question

write the quadratic equation whose roots are 5 and -6, and whose leading coefficient is 2.
(use the letter \\(x\\) to represent the variable.)

\\\box = 0\\

Explanation:

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<reasoning>

Set up the factored form of the quadratic equation

\[
a(x - r_1)(x - r_2) = 0
\]
\[
a = 2,\quad r_1 = 5,\quad r_2 = -6
\]

Substitute the given values into the equation

\[
2(x - 5)(x - (-6)) = 0
\]
\[
2(x - 5)(x + 6) = 0
\]

Expand and simplify the expression

\[
2(x^2 + 6x - 5x - 30) = 0
\]
\[
2(x^2 + x - 30) = 0
\]
\[
2x^2 + 2x - 60 = 0
\]
</reasoning>

<answer>
Write the quadratic equation whose roots are 5 and \(-6\), and whose leading coefficient is 2.

(Use the letter \(x\) to represent the variable.)

<blank>\(2x^2+2x-60\)</blank> \(= 0\)
</answer>

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"Mathematics",
"Algebra",
"Quadratic Equation Construction"
]
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</post_analysis>

Answer:

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<reasoning>

Set up the factored form of the quadratic equation

\[
a(x - r_1)(x - r_2) = 0
\]
\[
a = 2,\quad r_1 = 5,\quad r_2 = -6
\]

Substitute the given values into the equation

\[
2(x - 5)(x - (-6)) = 0
\]
\[
2(x - 5)(x + 6) = 0
\]

Expand and simplify the expression

\[
2(x^2 + 6x - 5x - 30) = 0
\]
\[
2(x^2 + x - 30) = 0
\]
\[
2x^2 + 2x - 60 = 0
\]
</reasoning>

<answer>
Write the quadratic equation whose roots are 5 and \(-6\), and whose leading coefficient is 2.

(Use the letter \(x\) to represent the variable.)

<blank>\(2x^2+2x-60\)</blank> \(= 0\)
</answer>

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"Mathematics",
"Algebra",
"Quadratic Equation Construction"
]
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