$-79\sqrt{3}$ is a root of $f(x) = x^2 - 18,723$ . 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

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-79\sqrt{3}

is a root of

f(x) = x^2 - 18,723

. 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.

Given root information: Since $-79\sqrt{3}$ is a root, the other root must also be $-79\sqrt{3}$ because the coefficients of the polynomial are real numbers, and non-real roots of polynomials with real coefficients always come in conjugate pairs.
Product of roots: To find the other root, we can use the fact that the product of the roots of a quadratic equation $ax^2 + bx + c = 0$ is $\frac{c}{a}$ . Here, $a = 1$ and $c = -18,723$ .
Calculate other root: The product of the roots is $(-79\sqrt{3}) \times (\text{other root}) = -18,723$ .
Simplify expression: Divide both sides by $-79\sqrt{3}$ to find the other root: (other root) = $\frac{-18,723}{-79\sqrt{3}}$ .
Rationalize denominator: Simplify the expression: $\text{(other root)} = \frac{18,723}{79\sqrt{3}}$ .
Calculate final answer: Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$ : (other root) = $\frac{18,723\sqrt{3}}{79 \times 3}$ .
Calculate final answer: Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$ : (other root) = $\frac{18,723\sqrt{3}}{79 \times 3}$ .Calculate the simplified value: (other root) = $\frac{18,723\sqrt{3}}{237}$ .
Calculate final answer: Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$ : (other root) = $\frac{18,723\sqrt{3}}{79 \times 3}$ .Calculate the simplified value: (other root) = $\frac{18,723\sqrt{3}}{237}$ .Divide $18,723$ by $237$ to get the final answer: (other root) = $79\sqrt{3}$ .