Let's analyze the given options to determine which system of equations can be used to find the roots of the equation [tex]\(12 x^3 - 5 x = 2 x^2 + x + 6\)[/tex].
To find the roots of the equation [tex]\(12 x^3 - 5 x = 2 x^2 + x + 6\)[/tex], we can rewrite it in a form that equates two separate functions to the same variable [tex]\(y\)[/tex].
We want to represent the equation in the form of two equations that are equal to [tex]\(y\)[/tex]:
1. Let's express the first part: [tex]\(12 x^3 - 5 x\)[/tex]
2. Let's express the second part: [tex]\(2 x^2 + x + 6\)[/tex]
Thus, this system of equations can be written as:
[tex]\[
\begin{cases}
y = 12 x^3 - 5 x \\
y = 2 x^2 + x + 6
\end{cases}
\][/tex]
We now compare this to the given options:
1. [tex]\(\left\{\begin{array}{l}y=12 x^3-5 x \\ y=2 x^2+x+6\end{array}\right.\)[/tex]
This represents the exact functions we derived, so this is the correct option.
2. [tex]\(\left\{\begin{array}{l}y=12 x^3-5 x+6 \\ y=2 x^2+x\end{array}\right.\)[/tex]
This is not correct as the terms rearranged do not match the expressions we derived.
3. [tex]\(\left\{\begin{array}{l}y=12 x^3-2 x^2-6 x \\ y=6\end{array}\right.\)[/tex]
This is also incorrect because it misses terms and alters expressions.
4. [tex]\(\left\{\begin{array}{l}y=12 x^3-2 x^2-6 x-6 \\ y=0\end{array}\right.\)[/tex]
This represents the equation in another rearranged form not suitable for determining the roots in the required system form.
Therefore, the correct system of equations to find the roots of the given equation is:
[tex]\[
\left\{\begin{array}{l}
y = 12 x^3 - 5 x \\
y = 2 x^2 + x + 6
\end{array}\right.
\][/tex]
