$-73+98 i$ is a root of $f(x)=x^{2}+146 x+$ $14933$ . Find the other roots of $\mathrm{f}(x)$ . $\newline$ Write your answer as a list of simplified values separated by commas, if there is more than one value.

Full solution

-73+98 i

is a root of

f(x)=x^{2}+146 x+

14933

. Find the other roots of

\mathrm{f}(x)

\newline

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

Complex Roots Conjugate Pairs: Since the coefficients of the polynomial are real numbers, the complex roots of polynomials with real coefficients come in conjugate pairs. This means that if $-73 + 98i$ is a root, then its conjugate, $-73 - 98i$ , must also be a root.
Sum of Roots Quadratic Equation: To find the other root, we can use the fact that the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is equal to $-\frac{b}{a}$ . In this case, the sum of the roots is $-146$ .
Setting up Equation for Other Root: We already know one of the roots is $-73 + 98i$ , so we can set up the equation $-146 = (-73 + 98i) + \text{other\_root}$ to solve for the other root.
Calculating Other Root: Subtracting the known root from $-146$ gives us the other root: $\text{other\_root} = -146 - (-73 + 98i) = -146 + 73 - 98i = -73 - 98i$ .
Final Result: Therefore, the other root of the polynomial $f(x)$ is $-73 - 98i$ .