Shaniah has a small square garden plot. She plans to expand her
garden plot by doubling the length and decreasing the width by 2
ft. Doing this will make the area of the resulting rectangular plot
to be 96 sq. ft larger than the original square garden plot.
Translate the situation into mathematical statement and write the equation in standard form.

Let's define the situation step by step and translate it into a mathematical equation.

1. Let **x** represent the side length of the original square garden plot.

   - The area of the original square plot is:
   \[
   A_{\text{square}} = x^2
   \]

2. After the expansion, Shaniah plans to:
   - Double the length of the plot, so the new length is \(2x\).
   - Decrease the width by 2 feet, so the new width is \(x - 2\).

   Therefore, the area of the expanded rectangular plot is:
   \[
   A_{\text{rectangle}} = 2x(x - 2)
   \]

3. The new rectangular plot's area is said to be 96 square feet larger than the original square plot, which gives us the following relationship:
   \[
   A_{\text{rectangle}} = A_{\text{square}} + 96
   \]

4. Substituting the expressions for the areas into this equation:
   \[
   2x(x - 2) = x^2 + 96
   \]

5. Now, simplify the equation to put it in standard form:

   First, expand the left-hand side:
   \[
   2x(x - 2) = 2x^2 - 4x
   \]

   The equation becomes:
   \[
   2x^2 - 4x = x^2 + 96
   \]

6. Move all terms to one side to set the equation equal to zero:
   \[
   2x^2 - 4x - x^2 - 96 = 0
   \]

   Simplify:
   \[
   x^2 - 4x - 96 = 0
   \]

### Final Equation (Standard Form):
\[
x^2 - 4x - 96 = 0
\]

This is the standard form of the equation that represents the problem.

Would you like to go into more detail on solving this equation, or do you have any questions?

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Here are five related questions:
1. How do you solve quadratic equations like this one?
2. What is the significance of "standard form" in algebraic equations?
3. What does it mean when a quadratic equation has no real solutions?
4. How do changes in length and width affect the area of a rectangle?
5. How can you use quadratic equations in real-life situations?

**Tip:** Quadratic equations often have two solutions, but both might not always be practical for real-world problems. Always verify which solution makes sense for the context.

Shaniah has a small square garden plot. She plans to expand her garden plot by doubling the length and decreasing the width by 2 ft. Doing this will make the area of the resulting rectangular plot 96 sq. ft larger than the original square garden plot. Translate the situation into a mathematical statement and write the equation in standard form.

Learn how to set up and solve a quadratic equation for an expanded garden plot problem. Shaniah's square garden plot changes into a rectangle, and the resulting area increases by 96 sq. ft. This solution involves key algebraic concepts like quadratic equations and geometry, suitable for Grades 8-10.

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