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!