To find the system of equations that can be used to identify the roots of the equation [tex]\( 2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9 \)[/tex], we start by analyzing the given equation and understanding the steps involved in transforming it.
Here are the detailed steps:
1. Starting Equation:
[tex]\[
2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9
\][/tex]
2. Move all terms to one side of the equation:
We can rearrange the equation such that all terms are on one side of the equality.
[tex]\[
2x^3 + 4x^2 - x + 5 + 3x^2 - 4x - 9 = 0
\][/tex]
3. Combine like terms:
Now we simplify by combining the terms involving [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and constant terms:
[tex]\[
2x^3 + 4x^2 + 3x^2 - x - 4x + 5 - 9 = 0
\][/tex]
[tex]\[
2x^3 + (4 + 3)x^2 + (-1 - 4)x + (5 - 9) = 0
\][/tex]
[tex]\[
2x^3 + 7x^2 - 5x - 4 = 0
\][/tex]
4. Formulating the System of Equations:
To approach this using a system of equations, we need to create two equations that represent the left-hand and the right-hand side of the original equation separately.
The original equation can be split into:
[tex]\[
y = 2x^3 + 4x^2 - x + 5
\][/tex]
[tex]\[
y = -3x^2 + 4x + 9
\][/tex]
Thus, the system of equations that you can use to find the roots of the original equation is:
[tex]\[
\begin{array}{l}
y = 2x^3 + 4x^2 - x + 5 \\
y = -3x^2 + 4x + 9
\end{array}
\][/tex]
So the correct system of equations is:
[tex]\[
\begin{array}{l}
y = 2x^3 + 4x^2 - x + 5 \\
y = -3x^2 + 4x + 9
\end{array}
\][/tex]
And that matches option 3 from the list provided.
