To solve the system of equations:
[tex]\[
\begin{cases}
y = 2x^2 + 6x + 4 \\
y = -4x^2 + 4
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where the two equations intersect. This requires setting the two equations equal to each other since both are equal to [tex]\( y \)[/tex].
1. Set the equations equal to each other:
[tex]\[
2x^2 + 6x + 4 = -4x^2 + 4
\][/tex]
2. Combine like terms and simplify the equation:
[tex]\[
2x^2 + 6x + 4 + 4x^2 - 4 = 0
\][/tex]
[tex]\[
6x^2 + 6x = 0
\][/tex]
3. Factor the equation:
[tex]\[
6x(x + 1) = 0
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
6x = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
[tex]\[
x = 0 \quad \text{or} \quad x = -1
\][/tex]
Now that we have the [tex]\( x \)[/tex]-values, we need to find the corresponding [tex]\( y \)[/tex]-values by substituting [tex]\( x = 0 \)[/tex] and [tex]\( x = -1 \)[/tex] back into either of the original equations.
5. Substitute [tex]\( x = 0 \)[/tex] into [tex]\( y = 2x^2 + 6x + 4 \)[/tex]:
[tex]\[
y = 2(0)^2 + 6(0) + 4
\][/tex]
[tex]\[
y = 4
\][/tex]
6. Substitute [tex]\( x = -1 \)[/tex] into [tex]\( y = 2x^2 + 6x + 4 \)[/tex]:
[tex]\[
y = 2(-1)^2 + 6(-1) + 4
\][/tex]
[tex]\[
y = 2(1) - 6 + 4
\][/tex]
[tex]\[
y = 2 - 6 + 4
\][/tex]
[tex]\[
y = 0
\][/tex]
Thus, the system of equations has two solutions:
1. [tex]\( (x, y) = (0, 4) \)[/tex]
2. [tex]\( (x, y) = (-1, 0) \)[/tex]
So, the points of intersection of the given system of equations are:
[tex]\[
(-1, 0) \quad \text{and} \quad (0, 4)
\][/tex]
