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$-67\sqrt{7}$ is a root of $f(x) = x^2 - 31,423$ . 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.

-67\sqrt{7}

is a root of

f(x) = x^2 - 31,423

. 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 Theorem: Since $-67\sqrt{7}$ is a root, by the Conjugate Root Theorem, the other root must be its conjugate, which is $67\sqrt{7}$ .
Calculate Product: To check, 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 $c = -31,423$ .
Verify Product Calculation: The product of the roots is $(-67\sqrt{7}) \times (67\sqrt{7}) = -67^2 \times 7 = -31,423$ .
Final Confirmation: Since the product of the roots equals $c$ , and we have found that the product of $-67\sqrt{7}$ and $67\sqrt{7}$ is indeed $-31,423$ , we have the correct second root.

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