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Let's go through the steps to solve the problem shown in the first image. The problem seems to be based on a rectangular flower bed and some unknown values that need to be found.

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### Image 1: Step-by-step breakdown:

**Step 1: Translate the situation into a quadratic equation.**

- We are given a rectangular flower bed with the following dimensions:
  - **Width of the flower bed**: 8 meters
  - **Length of the flower bed**: 10 meters
  
  There is a walking path (道) around the flower bed that is of a uniform width, and the total area of the flower bed plus the path is 48 square meters.

Let the unknown width of the path be represented by **x** meters. 

- The **total dimensions** of the flower bed plus the path will then be:
  - **Length**: \( 10 + 2x \) meters (since there is a path on both sides, the total length increases by \( 2x \)).
  - **Width**: \( 8 + 2x \) meters (similarly, the width increases by \( 2x \)).
  
The total area, including the path, is given as 48 square meters.

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**Step 2: Form a quadratic equation.**

The area of the flower bed plus the path is the product of the length and width, and we know it equals 48 square meters:

\[
(10 + 2x)(8 + 2x) = 48
\]

Now, expand the equation:

\[
(10 + 2x)(8 + 2x) = 80 + 20x + 16x + 4x^2 = 80 + 36x + 4x^2
\]

Set the equation equal to 48:

\[
80 + 36x + 4x^2 = 48
\]

Subtract 48 from both sides:

\[
4x^2 + 36x + 32 = 0
\]

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**Step 3: Solve the quadratic equation.**

We now have a quadratic equation:

\[
4x^2 + 36x + 32 = 0
\]

First, divide through by 4 to simplify:

\[
x^2 + 9x + 8 = 0
\]

Now, solve using the quadratic formula:

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

For this equation, \(a

A rectangular flower bed has a length of 10 meters and a width of 8 meters. There is a walking path of unknown width surrounding the bed, and the total area (including the path) is 48 square meters. Find the width of the path.

Learn how to solve for the width of a path surrounding a rectangular flower bed using quadratic equations. The problem involves determining the width of the path given the total area, length, and width of the bed. Suitable for Grades 8-10, this problem covers algebraic concepts and quadratic formulas.

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