Graph the following nonlinear system of equations and identify the two solutions.

y=x2+6x−4y=−x+4

Group of answer choices

(1,−3) and (8,−12)

(1,3) and (8,12)

(−1,3) and (−8,12)

(1,3) and (−8,12)

The graph shows the two equations:

1. \( y = x^2 + 6x - 4 \) (blue curve)
2. \( y = -x + 4 \) (red line)

From the graph, you can identify the points of intersection, which are the solutions to the system of equations. 

Now, let's determine the specific intersection points algebraically to confirm the solutions.

### Finding Solutions
To find the intersection points, set the two equations equal to each other:

\[
x^2 + 6x - 4 = -x + 4
\]

Rearranging gives:

\[
x^2 + 7x - 8 = 0
\]

Now, we can use the quadratic formula to solve for \( x \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = 1, b = 7, c = -8 \). Let's calculate the solutions for \( x \).The solutions to the system of equations are:

1. \( (1, 3) \)
2. \( (-8, 12) \)

### Answer Choices
From the options you provided, the correct pair of solutions is:
- **(1, 3) and (-8, 12)**

If you have any questions or need further details, feel free to ask!

### Related Questions
1. What are the steps to solve a nonlinear system of equations?
2. How does the quadratic formula work?
3. Can you explain how to identify intersection points graphically?
4. What are some applications of solving systems of equations?
5. How do you find the vertex of a quadratic function?

**Tip:** Always check your solutions by substituting them back into the original equations!

Graph the following nonlinear system of equations and identify the two solutions: y=x^2+6x−4, y=−x+4.

Discover how to graph and solve the nonlinear system of equations y=x^2+6x−4 and y=−x+4. This guide explains the intersection points and includes detailed steps using the quadratic formula, suitable for high school students.

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