Mr. Luna would like to construct a new house with a floor area of 72 m². Не asked an architect to prepare a floor plan that shows the following:
A. 2 bedrooms
B. Living room
C. Dining room
D. Comfort room
E. Kitchen
F. Laundry area

1. Suppose you were the architect asked by Mr. Luna to prepare a floor plan. How will you do it? Draw the floor plan.
2. Formulate as many quadratic equations using the floor plan that Solve the equations using the quadratic formula.

To create a floor plan and formulate quadratic equations based on the given criteria, let’s break down the steps:

### 1. Designing the Floor Plan
Given the total area of 72 m², we need to allocate space for the following rooms:
- **2 Bedrooms**
- **Living Room**
- **Dining Room**
- **Comfort Room**
- **Kitchen**
- **Laundry Area**

Let's allocate approximate areas for each room. Here’s a potential breakdown:
- **Bedroom 1**: 15 m²
- **Bedroom 2**: 12 m²
- **Living Room**: 18 m²
- **Dining Room**: 10 m²
- **Comfort Room**: 5 m²
- **Kitchen**: 8 m²
- **Laundry Area**: 4 m²

Total: \(15 + 12 + 18 + 10 + 5 + 8 + 4 = 72 \, \text{m}^2\)

Now, let's describe how to draw a simple floor plan based on these areas:
- **Bedrooms (15 m² and 12 m²)** could be placed next to each other.
- **Living Room (18 m²)** could be centrally located.
- **Dining Room (10 m²)** could be adjacent to the kitchen.
- **Kitchen (8 m²)** and **Laundry Area (4 m²)** might be placed together for convenience.
- **Comfort Room (5 m²)** could be near the bedrooms.

Given this arrangement, let’s move to the quadratic equations.

### 2. Formulating Quadratic Equations

Quadratic equations can be formed based on different aspects of the rooms. Here are a few examples:

#### Example 1: Relationship between Length and Width of a Room
Suppose the length \(L\) and width \(W\) of the **Living Room** (area = 18 m²) are related by \( L = W + 2 \) (i.e., the length is 2 meters more than the width).

The area \(A\) is given by:
\[
A = L \times W = 18
\]
Substitute \(L = W + 2\):
\[
(W + 2) \times W = 18
\]
\[
W^2 + 2W - 18 = 0
\]
This is a quadratic equation. Solving it using the quadratic formula:
\[
W = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \(a = 1\), \(b = 2\), and \(c = -18\):
\[
W = \frac{-2 \pm \sqrt{2^2 + 4 \times 18}}{2 \times 1}
\]
\[
W = \frac{-2 \pm \sqrt{4 + 72}}{2}
\]
\[
W = \frac{-2 \pm \sqrt{76}}{2}
\]
\[
W = \frac{-2 \pm 2\sqrt{19}}{2}
\]
\[
W = -1 \pm \sqrt{19}
\]
We discard the negative root as width cannot be negative:
\[
W \approx \sqrt{19} - 1 \approx 3.36 \, \text{m}
\]
Then, the length \(L\) is:
\[
L = W + 2 \approx 5.36 \, \text{m}
\]

#### Example 2: Area of Bedroom
Assume one of the bedrooms (15 m²) has a length that is 3 meters more than twice its width. Let the width be \(W\) and the length \(L = 2W + 3\).

The area equation is:
\[
L \times W = 15
\]
Substitute \(L = 2W + 3\):
\[
(2W + 3) \times W = 15
\]
\[
2W^2 + 3W - 15 = 0
\]
Using the quadratic formula:
\[
W = \frac{-3 \pm \sqrt{3^2 + 4 \times 2 \times 15}}{4}
\]
\[
W = \frac{-3 \pm \sqrt{9 + 120}}{4}
\]
\[
W = \frac{-3 \pm \sqrt{129}}{4}
\]
Solving for \(W\) gives the width of the room, and \(L = 2W + 3\) gives the length.

These examples illustrate how quadratic equations can arise from the relationships between the dimensions of the rooms in the floor plan.

Would you like more details or have any questions? 

### Related Questions:
1. What other types of quadratic equations can be formed based on different rooms?
2. How would you determine the dimensions of each room if the total area was different?
3. Can the quadratic equations be solved graphically? If so, how?
4. How can the quadratic formula be used to solve non-physical roots in construction?
5. What adjustments could be made to the floor plan if the area constraint was altered?

**Tip**: Always verify the solutions of quadratic equations within the context of the problem, as not all roots may be physically meaningful.

Explore how to construct a house floor plan based on specified room areas. Learn to formulate quadratic equations to solve dimensions of rooms like bedrooms, living room, dining room, comfort room, kitchen, and laundry area. Suitable for Grades 7-9.

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