https://file.aiquickdraw.com/mathbot/file/1725541728505s27uyxsb.jpg

Let's break down and solve the two quadratic problems shown in the image.

### Problem 1: Rectangular Garden

We are given:
- **Area**: \( A = 84 \, \text{m}^2 \)
- **Perimeter**: \( P = 38 \, \text{m} \)

We need to find the **length** and **width** of the garden.

Let's use the following relationships:
1. **Area of a rectangle**: \( A = L \times W \)
2. **Perimeter of a rectangle**: \( P = 2L + 2W \)

Where:
- \( L \) is the length,
- \( W \) is the width.

### Step 1: Use the perimeter formula to express one variable in terms of the other.

\[
P = 2L + 2W = 38
\]
Simplify the equation:
\[
L + W = 19 \quad \Rightarrow \quad L = 19 - W
\]

### Step 2: Substitute this into the area equation.

The area is given as \( 84 \, \text{m}^2 \):
\[
L \times W = 84
\]
Substitute \( L = 19 - W \):
\[
(19 - W) \times W = 84
\]
Expand the equation:
\[
19W - W^2 = 84
\]
Rearrange to get a quadratic equation:
\[
W^2 - 19W + 84 = 0
\]

### Step 3: Solve the quadratic equation.

We can solve the quadratic using the quadratic formula:
\[
W = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For \( W^2 - 19W + 84 = 0 \), we have:
- \( a = 1 \)
- \( b = -19 \)
- \( c = 84 \)

Now, let's calculate the values.

### Problem 2: Car and Truck Speeds

We are given:
- The car travels 20 km/h faster than the truck.
- The car covers 350 km in 2 hours less than the truck takes to cover the same distance.

Let:
- \( v_t \) be the speed of the truck in km/h.
- \( v_c = v_t + 20 \) be the speed of the car in km/h.

The relationship between distance, speed, and time is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

For the car, the time taken is:
\[
t_c = \frac{350}{v_t + 20}
\]

For the truck, the time is:
\[
t_t = \frac{350}{v_t}
\]

We are told that the car takes 2 hours less than the truck, so:
\[
\frac{350}{v_t} - \frac{350}{v_t + 20} = 2
\]

Let's solve this equation step by step.

I will now proceed to solve both problems.### Solution to Problem 1: Rectangular Garden

The quadratic equation yields two possible solutions for the dimensions of the garden:
- **Width = 12 m**, **Length = 7 m**
- **Width = 7 m**, **Length = 12 m**

So, the dimensions of the rectangular garden are **12 meters by 7 meters**.

### Solution to Problem 2: Car and Truck Speeds

From the equation, we find:
- The **truck's speed** is \( 50 \, \text{km/h} \).
- The **car's speed** is \( 70 \, \text{km/h} \) (since it travels 20 km/h faster than the truck).

---

Let me know if you want further details or have any questions. Here are some related problems to consider:

1. If the area of the garden were increased to 120 m² with the same perimeter, what would be the new dimensions?
2. How long would it take the car to cover 420 km at the same speed of 70 km/h?
3. What if the truck's speed was 40 km/h? How would that affect the car's speed and time?
4. How much area would a garden have with dimensions of 15 m by 8 m?
5. If the car traveled 25 km/h faster than the truck, how would the equation change?

**Tip**: For quadratic equations, always check the discriminant to know if the solutions are real and distinct!

1. A rectangular garden has an area of 84 m² and a perimeter of 38 m. Find its length and width. 2. A car travels 20 kph faster than a truck. The car covers 350 km in two hours less than the time it takes the truck to travel the same distance. What is the speed of the car? How about the truck?

Learn how to solve a quadratic equation involving the area and perimeter of a rectangular garden and the speed-time relationship of a car and truck. This problem covers key algebra concepts such as quadratic equations, area, and perimeter formulas, as well as speed, distance, and time relationships. Suitable for Grades 9-10 students.

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