$62\sqrt{3}$ is a root of $f(x) = x^2 - 11,532$ . 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.

Full solution

62\sqrt{3}

is a root of

f(x) = x^2 - 11,532

. 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.

Given root symmetry: Since $62\sqrt{3}$ is a root, the other root will also be $62\sqrt{3}$ because the roots of a quadratic equation are symmetric with respect to the y-axis when the coefficient of $x$ is $0$ .
Product of roots: To find the other root, we 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 $c = -11,532$ .
Calculate product: The product of the roots is $(62\sqrt{3}) \times \text{other\_root} = -11,532$ .
Solve for other root: Solve for the other root: $\text{other\_root} = \frac{-11,532}{62\sqrt{3}}$ .
Simplify other root: Simplify the other root: $\text{other\_root} = \frac{-11,532}{62\sqrt{3}} \times \left(\frac{\sqrt{3}}{\sqrt{3}}\right) = \frac{-11,532\sqrt{3}}{62 \times 3}$ .
Further simplify: Further simplify: $other\_root = \frac{-11,532\sqrt{3}}{186}$ .
Final other root: Divide to get the other root: $\text{other\_root} = -62\sqrt{3}$ .