The graph of this system of equations is used to solve [tex]4x^2 - 3x + 6 = 2x^4 - 9x^3 + 2x[/tex].

[tex]\[
\begin{cases}
y = 4x^2 - 3x + 6 \\
y = 2x^4 - 9x^3 + 2x
\end{cases}
\][/tex]

What represents the solution set?

A. [tex]$y$[/tex]-intercepts of the graph
B. [tex]$x$[/tex]-intercepts of the graph
C. [tex]$y$[/tex]-coordinates of the intersection points
D. [tex]$x$[/tex]-coordinates of the intersection points



Answer :

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]