To find the solution set of the system of equations:
[tex]\[
\left\{
\begin{array}{l}
y = 4 x^2 - 3 x + 6 \\
y = 2 x^4 - 9 x^3 + 2 x
\end{array}
\right.
\][/tex]
we need to determine the values of [tex]\( x \)[/tex] at which the equations are equal. In other words, we solve for [tex]\( x \)[/tex] such that:
[tex]\[ 4 x^2 - 3 x + 6 = 2 x^4 - 9 x^3 + 2 x \][/tex]
Step-by-step solution:
1. Set the equations equal to each other:
[tex]\[ 4 x^2 - 3 x + 6 = 2 x^4 - 9 x^3 + 2 x \][/tex]
2. Rearrange the equation to set it to zero:
[tex]\[ 0 = 2 x^4 - 9 x^3 + 2 x - 4 x^2 + 3 x - 6 \][/tex]
This simplifies to:
[tex]\[ 2 x^4 - 9 x^3 - 4 x^2 + 5 x - 6 = 0 \][/tex]
3. Solve for the values of [tex]\( x \)[/tex]:
Solving this polynomial equation for its roots (the values of [tex]\( x \)[/tex]) will give us the [tex]\( x \)[/tex]-coordinates of the points where the two curves intersect.
The roots of this polynomial equation represent the [tex]\( x \)[/tex]-values where the two curves intersect. Therefore, the solution set represents the [tex]\( x \)[/tex]-coordinates of the intersection points.
To answer the specific given question, the solution set represents:
[tex]\[ \boxed{x \text{-coordinates of the intersection points}} \][/tex]
