Certainly! Let's solve the given system of equations step-by-step.
Given the system of equations:
[tex]\[
\begin{cases}
y = 2x^2 + 6x + 4 \\
y = -4x^2 + 4
\end{cases}
\][/tex]
To find the solutions, we need to set the two equations equal to each other because they both equal [tex]\(y\)[/tex]. This will help us find the values of [tex]\(x\)[/tex].
1. Set the two equations equal to each other:
[tex]\[
2x^2 + 6x + 4 = -4x^2 + 4
\][/tex]
2. Combine like terms:
To solve for [tex]\(x\)[/tex], first move all terms to one side of the equation:
[tex]\[
2x^2 + 6x + 4 + 4x^2 - 4 = 0
\][/tex]
Simplify:
[tex]\[
6x^2 + 6x = 0
\][/tex]
3. Factor the equation:
Factor out the common term:
[tex]\[
6x (x + 1) = 0
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Set each factor equal to zero:
[tex]\[
6x = 0 \quad \Rightarrow \quad x = 0
\][/tex]
[tex]\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\][/tex]
We have two solutions for [tex]\(x\)[/tex]: [tex]\(x = 0\)[/tex] and [tex]\(x = -1\)[/tex].
5. Substitute [tex]\(x\)[/tex] back into one of the original equations:
Let's use [tex]\(y = 2x^2 + 6x + 4\)[/tex] for substitution.
- For [tex]\(x = 0\)[/tex]:
[tex]\[
y = 2(0)^2 + 6(0) + 4 = 4
\][/tex]
- For [tex]\(x = -1\)[/tex]:
[tex]\[
y = 2(-1)^2 + 6(-1) + 4 = 2(1) - 6 + 4 = 2 - 6 + 4 = 0
\][/tex]
6. Write the solutions as ordered pairs [tex]\((x,y)\)[/tex]:
[tex]\[
(0, 4) \quad \text{and} \quad (-1, 0)
\][/tex]
Therefore, the solutions to the system of equations are:
[tex]\[
\boxed{(0, 4) \ \text{and} \ (-1, 0)}
\][/tex]
