Aisha wants to make two quilts, each with the same area. The first quilt will be square with sides s feet long. The second quilt will be a rectangle with a width that is half the length of a side of the square quilt and a length that is 6 feet longer than a side length of the square quilt.

Which quadratic equation can be used to find s, the side length of the square quilt?

s2 = (s + 6)
s2 = (s)(s + 6)
s2 = (6s)
s2 = (s)(6s)

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.

Aisha wants to make two quilts, each with the same area. The first quilt will be square with sides s feet long. The second quilt will be a rectangle with a width that is half the length of a side of the square quilt and a length that is 6 feet longer than a side length of the square quilt. Which quadratic equation can be used to find s, the side length of the square quilt?

Explore how to solve a quadratic equation to find the side length of a square quilt where the areas of the square and a rectangle (with a width half the square's side and a length 6 feet longer) are equal. This detailed solution covers key algebraic and geometric concepts, ideal for students in Grades 9-10.

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