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$-57\sqrt{6}$ is a root of $f(x) = x^2 - 19,494$ . Find the other roots of $f(x)$ . $\newline$ Write your answer as a list of simplified values separated by commas, if there is more than one value.

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Conjugate root theorems

Full solution

Q.

-57\sqrt{6}

is a root of

f(x) = x^2 - 19,494

. Find the other roots of

f(x)

.

\newline

Write your answer as a list of simplified values separated by commas, if there is more than one value.

Identify Root Relationship: Since $-57\sqrt{6}$ is a root, the other root will also be the negative of it because the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is $-b/a$ . In this case, $a = 1$ and $b = 0$ , so the sum of the roots is $0$ .
Calculate Other Root: The other root is therefore $57\sqrt{6}$ , because $-57\sqrt{6} + 57\sqrt{6} = 0$ .
List Roots: List the roots: $-57\sqrt{6}$ , $57\sqrt{6}$ .

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