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$88$ is a root of $f(x) = x^2 + 7,744$ . 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. $\newline$ ______

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Complex conjugate theorem

Full solution

Q.

88

is a root of

f(x) = x^2 + 7,744

. 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.

\newline

______

Identify root as $x - 88$ : Since $88$ is a root, we can write it as $x - 88 = 0$ . Now we need to find the other root.
Find other root: The polynomial is $f(x) = x^2 + 7,744$ . Since we know one root is $88$ , we can use the fact that the sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is $-\frac{b}{a}$ . But here, $b = 0$ since there is no $x$ term, so the other root is just $-88$ .
Verify roots in polynomial: Now we check if $88$ and $-88$ are indeed the roots by plugging them into the polynomial. For $x = 88$ , $f(88) = 88^2 + 7,744 = 7,744 + 7,744$ , which is true since $88^2 = 7,744$ . For $x = -88$ , $f(-88) = (-88)^2 + 7,744 = 7,744 + 7,744$ , which is also true.

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