https://file.aiquickdraw.com/mathbot/file/17277468907198g2cobje.jpg

Let's break down the problem in the image you provided:

The problem states:

- If we increase one side of a square by 3 units and decrease the other side by 1 unit, the area of the resulting rectangle would be 21 square units greater than the area of the original square. You are asked to find the length of the sides of the original square.

Let the side length of the original square be \( x \).

### Steps to Solve:
1. **Area of the original square**: The area of the square is \( x^2 \).

2. **Area of the new rectangle**: The dimensions of the new rectangle are:
   - One side is increased by 3 units, making it \( x + 3 \).
   - The other side is decreased by 1 unit, making it \( x - 1 \).

   So, the area of the new rectangle is \( (x + 3)(x - 1) \).

3. **Equation setup**: 
   The area of the rectangle is 21 units greater than the area of the original square. Therefore, we have the equation:

   \[
   (x + 3)(x - 1) = x^2 + 21
   \]

4. **Simplify the equation**:
   Expanding the left side:
   \[
   (x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3
   \]
   So, the equation becomes:
   \[
   x^2 + 2x - 3 = x^2 + 21
   \]

5. **Solve for \( x \)**:
   Subtract \( x^2 \) from both sides:
   \[
   2x - 3 = 21
   \]
   Add 3 to both sides:
   \[
   2x = 24
   \]
   Divide by 2:
   \[
   x = 12
   \]

So, the side length of the original square is **12 units**.

Would you like further clarification or details on any part of this solution?

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Here are 5 related questions you might find useful:
1. How do you factor quadratic equations like \( (x + 3)(x - 1) \)?
2. What are the properties of a square compared to a rectangle?
3. How do you solve more complex quadratic equations?
4. How can we determine the dimensions of a rectangle given its area?
5. What is the geometric significance of increasing and decreasing dimensions in such problems?

**Tip**: When expanding binomials, always check if you can factor and simplify the equation to make solving easier!

If we increased one side of a square by 3 units and decreased the other side by 1 unit, the area of the resulting rectangle would be 21 square units greater than the area of the square. How long are the sides of the original square?

Learn how to solve a quadratic equation involving the area of a square and a rectangle where one side is increased by 3 units and another decreased by 1 unit. This step-by-step solution will help students in Grades 8-10 understand how to relate algebra to geometry.

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