16. What quadratic equation can be formed if the two roots are -3 and-4?

x ^ 2 + x + 12 = 0

x ^ 2 + 7x + 12 = 0

x ^ 2 + x - 12 = 0

x ^ 2 - 7x + 12 = 0

17. The length of the garden is 7m longer than its width and the area is 18m ^ 2 If the width is represented as x, what equation can be formed from the situation?

x + (x + 7) = 14

x + (x - 7) = 14

x(x - 7) = 14

x(x + 7) = 14

18. The length of the garden is 7m longer than its width and the area is 18m ^ 2 How long is the garden?

2m

5m

7m

9m

19. The length of a rectangular playground is 25 meters more than its width and the area is 3000 square meters. What is the correct equation to represent the problem?

w(w - 25) > 3000

(w - 3000)(w + 25) = 0

w(w + 25) = 3000

(w + 25)(w - 25) = 3000

20. The length of a rectangular playground is 25 meters more than its width and the area of the pool is 3000 square meters.

What are the dimensions of the playground?

a. 20m by 150m

c. 40m by 75m

b. 24m by 125m

d. 50 m by 60 m

Let's solve these questions one by one.

### **16. What quadratic equation can be formed if the two roots are -3 and -4?**

If the roots of a quadratic equation are \( r_1 \) and \( r_2 \), the equation can be written as:
\[
(x - r_1)(x - r_2) = 0
\]
Given the roots \( r_1 = -3 \) and \( r_2 = -4 \), the quadratic equation becomes:
\[
(x + 3)(x + 4) = 0
\]
Expanding this, we get:
\[
x^2 + 7x + 12 = 0
\]
**Answer: \( x^2 + 7x + 12 = 0 \)**

### **17. The length of the garden is 7m longer than its width, and the area is \( 18 \, m^2 \). If the width is represented as \( x \), what equation can be formed from the situation?**

Let the width be \( x \) meters. Then the length is \( x + 7 \) meters. The area \( A \) of the garden is given by:
\[
A = \text{length} \times \text{width} = x(x + 7)
\]
Given that the area is \( 18 \, m^2 \):
\[
x(x + 7) = 18
\]
**Answer: \( x(x + 7) = 18 \)**

### **18. The length of the garden is 7m longer than its width and the area is \( 18 \, m^2 \). How long is the garden?**

From the equation \( x(x + 7) = 18 \), we solve for \( x \):
\[
x^2 + 7x - 18 = 0
\]
We solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 7 \), and \( c = -18 \):
\[
x = \frac{-7 \pm \sqrt{7^2 + 72}}{2}
\]
\[
x = \frac{-7 \pm \sqrt{121}}{2}
\]
\[
x = \frac{-7 \pm 11}{2}
\]
This gives us two possible values for \( x \):
\[
x = 2 \quad \text{or} \quad x = -9 \quad (\text{which is not physically meaningful for a length})
\]
So, the width \( x = 2 \), and the length \( x + 7 = 9 \) meters.

**Answer: 9m**

### **19. The length of a rectangular playground is 25 meters more than its width, and the area is \( 3000 \, m^2 \). What is the correct equation to represent the problem?**

Let the width be \( w \) meters. Then the length is \( w + 25 \) meters. The area \( A \) is given by:
\[
A = \text{length} \times \text{width} = w(w + 25)
\]
Given that the area is \( 3000 \, m^2 \):
\[
w(w + 25) = 3000
\]
**Answer: \( w(w + 25) = 3000 \)**

### **20. The length of a rectangular playground is 25 meters more than its width, and the area is \( 3000 \, m^2 \). What are the dimensions of the playground?**

We already have the equation:
\[
w(w + 25) = 3000
\]
Let's solve for \( w \):
\[
w^2 + 25w - 3000 = 0
\]
Using the quadratic formula:
\[
w = \frac{-25 \pm \sqrt{25^2 + 12000}}{2}
\]
\[
w = \frac{-25 \pm \sqrt{625 + 12000}}{2}
\]
\[
w = \frac{-25 \pm \sqrt{12625}}{2}
\]
\[
w \approx \frac{-25 \pm 112.36}{2}
\]
This gives us two possible values for \( w \):
\[
w \approx 43.68 \quad \text{or} \quad w \approx -68.68 \quad (\text{not physically meaningful})
\]
Taking \( w \approx 43.68 \) meters, the length is \( w + 25 \approx 68.68 \) meters. The closest answer to this is \( 40m \) by \( 75m \).

**Answer: \( 40m \) by \( 75m \)**

---

Do you want any more details or explanations?

Here are five related questions:

1. How do you derive the quadratic formula from a general quadratic equation?
2. What happens if a quadratic equation has a discriminant less than zero?
3. Can you explain why negative roots might be discarded in physical problems?
4. How do you solve a quadratic equation by completing the square?
5. What are the general steps to factor a quadratic equation?

**Tip:** When solving word problems involving quadratic equations, carefully define your variables and translate the problem's conditions into an equation step by step.

16. What quadratic equation can be formed if the two roots are -3 and -4? 

17. The length of the garden is 7m longer than its width and the area is 18m^2. If the width is represented as x, what equation can be formed from the situation? 

18. The length of the garden is 7m longer than its width and the area is 18m^2. How long is the garden? 

19. The length of a rectangular playground is 25 meters more than its width and the area is 3000 square meters. What is the correct equation to represent the problem? 

20. The length of a rectangular playground is 25 meters more than its width and the area of the pool is 3000 square meters. What are the dimensions of the playground?

Explore how to form and solve quadratic equations from real-world problems, including finding the dimensions of a garden and playground with given areas. This guide covers key algebraic concepts such as the quadratic formula, factoring, and applying the zero product property. Ideal for students in Grades 8-10.

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