To determine which system of equations can be used to find the roots of the equation [tex]\( 4x^2 = x^3 + 2x \)[/tex], let's first recall how systems of equations work for finding roots of a given equation. We need two equations that both use a common [tex]\( y \)[/tex] value.
1. Given the equation [tex]\( 4x^2 = x^3 + 2x \)[/tex], we will set one function equal to [tex]\( y \)[/tex]:
[tex]\[
y = 4x^2
\][/tex]
2. Similarly, set the other side of the original equation equal to [tex]\( y \)[/tex]:
[tex]\[
y = x^3 + 2x
\][/tex]
We now have a system of equations:
[tex]\[
\begin{cases}
y = 4x^2 \\
y = x^3 + 2x
\end{cases}
\][/tex]
This system of equations is used to find the points of intersection between the parabola [tex]\( y = 4x^2 \)[/tex] and the cubic function [tex]\( y = x^3 + 2x \)[/tex]. Solving this system will give us the [tex]\( x \)[/tex]-values (roots) for which the equation [tex]\( 4x^2 = x^3 + 2x \)[/tex] holds true.
Reviewing the given options:
1. [tex]\(\begin{cases} y = -4x^2 \\ y = x^3 + 2x \end{cases}\)[/tex]:
- Incorrect, since [tex]\( y = -4x^2 \)[/tex] is not derived from the original equation.
2. [tex]\(\begin{cases} y = x^3 - 4x^2 + 2x \\ y = 0 \end{cases}\)[/tex]:
- Incorrect, because solving [tex]\( 4x^2 - (x^3 + 2x) = 0 \)[/tex] is not explicitly represented here.
3. [tex]\(\begin{cases} y = 4x^2 \\ y = -x^3 - 2x \end{cases}\)[/tex]:
- Incorrect, because [tex]\( y = -x^3 - 2x \)[/tex] does not match the original equation.
4. [tex]\(\begin{cases} y = 4x^2 \\ y = x^3 + 2x \end{cases}\)[/tex]:
- This is correct because both expressions match the set [tex]\( 4x^2 = x^3 + 2x \)[/tex].
Thus, the correct system of equations that can be used to find the roots of the equation [tex]\( 4x^2 = x^3 + 2x \)[/tex] is:
[tex]\[
\begin{cases}
y = 4x^2 \\
y = x^3 + 2x
\end{cases}
\][/tex]
