To find the roots of the equation [tex]\(4x^2 = x^3 + 2x\)[/tex], we need to express it as a system of equations. This system of equations is useful because the intersection points of the two equations will give us the roots of the original equation.
Let’s break down the process:
1. Original Equation:
[tex]\[ 4x^2 = x^3 + 2x \][/tex]
2. System of Equations Approach:
We can think of the left side of the equation [tex]\(4x^2\)[/tex] as one function [tex]\(y\)[/tex], and the right side of the equation [tex]\(x^3 + 2x\)[/tex] as another function [tex]\(y\)[/tex].
Essentially, we want:
[tex]\[ y = 4x^2 \][/tex]
And
[tex]\[ y = x^3 + 2x \][/tex]
3. Form the System of Equations:
Now we put these two functions together in one system of equations:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 \\
y = x^3 + 2x
\end{array}
\right.
\][/tex]
Thus, the required system of equations to find the roots of the equation [tex]\(4x^2 = x^3 + 2x\)[/tex] is:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 \\
y = x^3 + 2x
\end{array}
\right.
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 \\
y = x^3 + 2x
\end{array}
\right.
\][/tex]
