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$-59\sqrt{2}$ is a root of $f(x) = x^2 - 6,962$ . 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.

-59\sqrt{2}

is a root of

f(x) = x^2 - 6,962

. 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.

Root Symmetry Explanation: Since $-59\sqrt{2}$ is a root, the other root will also be $-59\sqrt{2}$ because the roots of a quadratic equation are symmetric with respect to the y-axis when the coefficient of $x$ is $0$ .
Sum of Roots Formula: To find the other root, we can use the sum of roots formula for a quadratic equation $ax^2 + bx + c = 0$ , which is $-\frac{b}{a}$ . However, since $b = 0$ in our equation, the sum of the roots is $0$ .
Calculation of Other Root: So, if one root is $-59\sqrt{2}$ , the other root must be $59\sqrt{2}$ to make the sum of the roots equal to $0$ .
Final Roots of the Polynomial: Therefore, the roots of the polynomial $f(x) = x^2 - 6,962$ are $-59\sqrt{2}$ and $59\sqrt{2}$ .

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