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$79\sqrt{7}$ is a root of $f(x) = x^2 - 43,687$ . 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.

79\sqrt{7}

is a root of

f(x) = x^2 - 43,687

. 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 Roots: Since $79\sqrt{7}$ is a root, the other root will be its opposite because the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is $-b/a$ , and here $a = 1$ and $b = 0$ . So the other root is $-79\sqrt{7}$ .
Check Roots Using Formula: Now, let's check the roots by using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Since $b = 0$ , the formula simplifies to $x = \pm\frac{\sqrt{-4ac}}{2a}$ . For our equation, $a = 1$ and $c = -43,687$ , so we substitute these values in.
Calculate Discriminant: Calculating the discriminant: $\sqrt{b^2 - 4ac} = \sqrt{0 - 4(1)(-43,687)} = \sqrt{174748}.$
Simplify Discriminant: Simplifying the discriminant: $\sqrt{174748} = 418\sqrt{1} = 418$ , because the square root of $1$ is $1$ .

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