https://file.aiquickdraw.com/mathbot/file/1726026917438n245p5gw.jpg

The equation provided is:

\[
x^2 + 4x - 12 = |x + 4| + 3
\]

To approximate the solutions of this equation using graphing technology, we can graph the two sides of the equation separately and look for their points of intersection. Here’s the step-by-step process:

1. **Graph the Left Side**: 
   The left side of the equation is a quadratic function:
   \[
   f(x) = x^2 + 4x - 12
   \]

2. **Graph the Right Side**:
   The right side of the equation involves an absolute value:
   \[
   g(x) = |x + 4| + 3
   \]

3. **Find Points of Intersection**:
   The points where the graphs of \(f(x)\) and \(g(x)\) intersect are the solutions to the equation. These are the values of \(x\) that satisfy both expressions.

### Steps to Use a Graphing Calculator or Software:

1. **Input the Functions**:
   - Enter \(y_1 = x^2 + 4x - 12\)
   - Enter \(y_2 = |x + 4| + 3\)

2. **Set the Viewing Window**:
   Adjust the viewing window to a range that shows the behavior of both functions clearly. A suggested range is:
   - \(x\)-axis: \([-10, 10]\)
   - \(y\)-axis: \([-20, 20]\)

3. **Find the Intersections**:
   Use the "intersect" feature on your graphing calculator or software to find the approximate points where the graphs intersect.

### Approximation Without Technology:

Without graphing, we can analyze the functions:
- The quadratic function \(x^2 + 4x - 12\) is a parabola opening upwards.
- The absolute value function \(|x + 4| + 3\) forms a V-shape.

By solving graphically or through numerical methods, we can approximate the points where these two graphs intersect.

Would you like me to provide a specific graphical solution, or do you need help with any further details?

### Related Questions:

1. What is the significance of the absolute value in the equation?
2. How can you solve the equation algebraically by considering different cases for the absolute value?
3. What are the properties of the functions involved that affect the number of solutions?
4. Can you find the vertex and axis of symmetry for the quadratic function?
5. How does the graph of \(|x + 4|\) shift compared to \(|x|\)?

**Tip**: Remember that absolute value functions create "V" shapes on graphs, which can intersect quadratic curves in 0, 1, or 2 points depending on their relative positions.

Use graphing technology to approximate the solutions of the equation: x^2 + 4x - 12 = |x + 4| + 3

Learn how to solve the equation x^2 + 4x - 12 = |x + 4| + 3 using graphing technology to find the points of intersection. This problem involves key algebraic concepts such as quadratic equations and absolute value functions. Suitable for high school students in Grades 9-12.

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