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 analyze each problem step by step.

### 16. **Quadratic equation with roots -3 and -4**

To form a quadratic equation from the roots, you use the fact that if \( r_1 \) and \( r_2 \) are the roots, the equation can be written as:

\[
(x - r_1)(x - r_2) = 0
\]

Here, \( r_1 = -3 \) and \( r_2 = -4 \). Substituting these into the equation:

\[
(x + 3)(x + 4) = 0
\]

Expanding this:

\[
x^2 + 4x + 3x + 12 = 0
\]

Simplifying:

\[
x^2 + 7x + 12 = 0
\]

**Correct answer: \(\mathbf{x^2 + 7x + 12 = 0}\)**

### 17. **Equation representing the garden dimensions**

Let the width be \( x \), and the length is 7 meters longer, so the length is \( x + 7 \). The area of the garden is given as 18 square meters. The area of a rectangle is given by:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

Substituting the values:

\[
x(x + 7) = 18
\]

**Correct answer: \(\mathbf{x(x + 7) = 18}\)**

### 18. **Finding the length of the garden**

Using the equation \( x(x + 7) = 18 \), we solve for \( x \) (the width).

\[
x^2 + 7x - 18 = 0
\]

This quadratic equation can be solved by factoring or using the quadratic formula:

\[
x = \frac{-7 \pm \sqrt{7^2 + 4 \times 18}}{2 \times 1}
\]

The width \( x = 2 \) meters, so the length is:

\[
x + 7 = 2 + 7 = 9 \text{ meters}
\]

**Correct answer: \(\mathbf{9 \text{ meters}}\)**

### 19. **Equation for the playground dimensions**

Let the width be \( w \), and the length is 25 meters longer, so the length is \( w + 25 \). The area of the playground is 3000 square meters. The equation becomes:

\[
w(w + 25) = 3000
\]

**Correct answer: \(\mathbf{w(w + 25) = 3000}\)**

### 20. **Finding the dimensions of the playground**

Using the equation \( w(w + 25) = 3000 \), solve for \( w \):

\[
w^2 + 25w - 3000 = 0
\]

Factoring or using the quadratic formula:

\[
w = \frac{-25 \pm \sqrt{25^2 + 4 \times 3000}}{2 \times 1}
\]

The correct solution gives \( w = 40 \) meters. The length is \( w + 25 = 65 \) meters.

**Correct answer: \(\mathbf{40 \text{ meters by } 75 \text{ meters}}\)**

---

Would you like more detailed steps or have any other questions? Here are some related questions to consider:

1. How do you derive the quadratic formula?
2. What are some methods for solving quadratic equations?
3. Can the dimensions of a rectangle always be determined from the area alone?
4. How can you confirm that a given quadratic equation has real roots?
5. What are the key properties of a quadratic equation?

**Tip:** For quadratic equations, always check if they can be factored easily before using the quadratic formula.

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 solving quadratic equations with roots -3 and -4, and understand how to determine the dimensions of a garden and playground using quadratic and area formulas. This content is suitable for students in Grades 8-10.

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