Let's break down the problem:

Step 1: Area of the Square Quilt

The area of the square quilt is given by: $\text{Area of square} = s^2$ where $s$ is the side length of the square quilt.

Step 2: Area of the Rectangular Quilt

The rectangular quilt has:

• Width = half the length of a side of the square quilt, i.e., $\frac{s}{2}$,
• Length = 6 feet longer than the side of the square quilt, i.e., $s + 6$.

So, the area of the rectangular quilt is: $\text{Area of rectangle} = \text{Width} \times \text{Length} = \left( \frac{s}{2} \right)(s + 6)$

Step 3: Setting the Areas Equal

Since the two quilts have the same area, we set the areas equal: $s^2 = \left( \frac{s}{2} \right)(s + 6)$

Step 4: Simplifying the Equation

We can now simplify the equation: $s^2 = \frac{s(s + 6)}{2}$ Multiply both sides by 2 to eliminate the fraction: $2s^2 = s(s + 6)$ Distribute the $s$ on the right side: $2s^2 = s^2 + 6s$

Step 5: Forming the Quadratic Equation

Subtract $s^2 + 6s$ from both sides: $2s^2 - s^2 - 6s = 0$ Simplify: $s^2 - 6s = 0$

This is the quadratic equation that can be used to find $s$.

Conclusion:

The correct equation is not one of the provided options directly, but the equation we derived is: $s^2 - 6s = 0$

Do you need more details or have any questions?

Here are 5 related questions:

1. How can we solve $s^2 - 6s = 0$ to find the value of $s$?
2. How do you solve a quadratic equation using factoring?
3. What are the steps to check if the solutions to a quadratic equation are valid for a given problem?
4. What other forms can quadratic equations take?
5. How do different quadratic equations relate to geometrical shapes like squares and rectangles?

Tip: Always make sure the units (such as feet or meters) are consistent throughout a geometry problem to avoid errors.