$-55\sqrt{3}$ is a root of $f(x) = x^2 - 9,075$ . 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.

Full solution

-55\sqrt{3}

is a root of

f(x) = x^2 - 9,075

. 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.

Roots Conjugate Pairs: Since $-55\sqrt{3}$ is a root, the other root must also be $-55\sqrt{3}$ 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 $\frac{c}{a}$ . For $f(x) = x^2 - 9,075$ , $a = 1$ and $c = -9,075$ .
Finding Other Root: The product of the roots is $-9,075$ . So, if one root is $-55\sqrt{3}$ , the other root, let's call it $r$ , must satisfy $(-55\sqrt{3}) \times r = -9,075$ .
Solving for Other Root: Solving for $r$ , we get $r = -\frac{9,075}{-55\sqrt{3}}$ . Simplify this by multiplying the numerator and denominator by $\sqrt{3}$ to rationalize the denominator.
Simplify Fraction: $r = \frac{-9,075\sqrt{3}}{55 \times 3}$ . Now, simplify the fraction.
Final Result: $r = \frac{-9,075\sqrt{3}}{165}$ . Divide both numerator and denominator by $165$ .
Final Result: $r = \frac{-9,075\sqrt{3}}{165}$ . Divide both numerator and denominator by $165$ . $r = -55\sqrt{3}$ . So, the other root is also $-55\sqrt{3}$ .