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$-108\sqrt{5}$ is a root of $f(x) = x^2 - 58,320$ . 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.

-108\sqrt{5}

is a root of

f(x) = x^2 - 58,320

. 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 $-108\sqrt{5}$ is a root, the other root will also be the negative of this value 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 Second Root: The other root is therefore $108\sqrt{5}$ .
Verify Sum of Roots: Check the roots by adding them together: $-108\sqrt{5} + 108\sqrt{5} = 0$ , which confirms that the sum of the roots is indeed $0$ .
List Final Roots: List the roots: $-108\sqrt{5}$ , $108\sqrt{5}$ .

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