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$-146\sqrt{2}$ is a root of $f(x) = x^2 - 42,632$ . 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.

-146\sqrt{2}

is a root of

f(x) = x^2 - 42,632

. 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 $-146\sqrt{2}$ 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, $b = 0$ , so the sum of the roots is $0$ .
Calculate second root: The other root is $146\sqrt{2}$ . We can check this by adding the roots: $-146\sqrt{2} + 146\sqrt{2} = 0$ , which satisfies the condition that the sum of the roots is $0$ .
List all roots: List the roots: $-146\sqrt{2}$ , $146\sqrt{2}$ .

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