To find the roots of the given equation [tex]\( 12x^3 - 5x = 2x^2 + x + 6 \)[/tex], we first need to form an equation in standard polynomial form (i.e., set the equation to zero).
Step-by-Step Solution:
1. Rewrite the given equation:
[tex]\[
12x^3 - 5x = 2x^2 + x + 6
\][/tex]
2. Move all terms to one side of the equation:
[tex]\[
12x^3 - 5x - 2x^2 - x - 6 = 0
\][/tex]
3. Combine like terms:
[tex]\[
12x^3 - 2x^2 - 6x - 6 = 0
\][/tex]
4. Represent this polynomial equation with [tex]\(y\)[/tex]:
[tex]\[
y = 12x^3 - 2x^2 - 6x - 6
\][/tex]
To find the roots, we need to set [tex]\( y = 0 \)[/tex].
Thus, the system of equations that can be used to find the roots is:
[tex]\[
\left\{\begin{array}{l}
y = 12x^3 - 2x^2 - 6x - 6 \\
y = 0
\end{array}\right.
\][/tex]
The correct answer is:
[tex]\[
\left\{\begin{array}{l}
y = 12x^3 - 2x^2 - 6x - 6 \\
y = 0
\end{array}\right.
\][/tex]
