To solve the system of equations given by:
[tex]\[
\left\{
\begin{array}{l}
y = 4x^2 - 3x + 6 \\
y = 2x^4 - 9x^3 + 2x
\end{array}
\right.
\][/tex]
we want to find the points where these two curves intersect. This means finding the [tex]\(x\)[/tex]-coordinates where the two equations have the same value of [tex]\(y\)[/tex]. By setting the right-hand sides of the equations equal to each other, we get the equation:
[tex]\[
4x^2 - 3x + 6 = 2x^4 - 9x^3 + 2x
\][/tex]
We need to solve this equation for [tex]\(x \)[/tex]. After solving this, we obtain the [tex]\(x\)[/tex]-coordinates of the intersection points. The result provided gives us the [tex]\(x\)[/tex]-coordinates of the points where the two curves intersect:
[tex]\[
[0.40614865893028 + 0.613860190379217i, 4.83334625362885, 0.40614865893028 - 0.613860190379217i, -1.14564357148941]
\][/tex]
Interpreting these values, we observe that some of the solutions are complex numbers (indicated by the presence of [tex]\(i\)[/tex]), but the real solutions among them are:
- [tex]\( x \approx 4.83334625362885 \)[/tex]
- [tex]\( x \approx -1.14564357148941 \)[/tex]
Hence, the solution set represents the [tex]\(x\)[/tex]-coordinates of the intersection points of the two equations. Therefore, the correct answer is:
[tex]\[
\boxed{x\text{-coordinates of the intersection points}}
\][/tex]
