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In this problem, we are given the dimensions of an open box, and the task is to write a quadratic equation to find the unknown dimension \( x \), given that the volume of the box is 182 cubic centimeters.

### Step 1: Understand the box dimensions
- The length of the base is \( x + 6 \).
- The width of the base is \( x \).
- The height of the box is 2 cm (as shown by the folded sides).

### Step 2: Write the formula for the volume of a rectangular box
The formula for the volume \( V \) of a rectangular box is:

\[
V = \text{length} \times \text{width} \times \text{height}
\]

Using the dimensions from the diagram:
- Length = \( x + 6 \)
- Width = \( x \)
- Height = \( 2 \)

The volume equation becomes:

\[
V = (x + 6) \cdot x \cdot 2
\]

### Step 3: Substitute the given volume
We are told that the volume is 182 cubic centimeters, so:

\[
(x + 6) \cdot x \cdot 2 = 182
\]

### Step 4: Simplify the equation
First, divide both sides by 2 to simplify:

\[
(x + 6) \cdot x = 91
\]

Now, expand the left-hand side:

\[
x(x + 6) = 91
\]
\[
x^2 + 6x = 91
\]

### Step 5: Form the quadratic equation
Rearrange the equation to standard quadratic form:

\[
x^2 + 6x - 91 = 0
\]

This is the quadratic equation that can be used to solve for \( x \).

Would you like details on solving this quadratic equation, or any other questions on this topic?

### 5 Related Questions:
1. How can you solve the quadratic equation \( x^2 + 6x - 91 = 0 \)?
2. What are the possible values of \( x \) once the quadratic equation is solved?
3. How does the box's volume change if the height is doubled?
4. What is the relationship between the volume and the surface area of the box?
5. Can you use the quadratic formula to solve this type of box dimension problem?

### Tip:
When expanding or simplifying expressions, always check for common factors to reduce the equation before solving. This helps avoid mistakes and simplifies the process!

Suppose that the volume of the box is 182 cubic centimeters. Use the given information to write a quadratic equation that can be used to find x.

Learn how to write and solve a quadratic equation to find the dimensions of a box given its volume. This problem covers key algebra concepts, including the formula for the volume of a rectangular prism and the steps to simplify equations into standard quadratic form. Suitable for students in Grades 8-10.

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