Josephine Has A Rectangular Garden With An Area Of 2 X 2 + X − 6 2x^2 + X - 6 2 X 2 + X − 6 Square Feet.Which Expressions Can Represent The Length And Width Of The Garden?A. Length = X 2 − 3 X^2 - 3 X 2 − 3 Feet; Width = 2 2 2 Feet B. Length = 2 X + 3 2x + 3 2 X + 3

Introduction

In mathematics, the area of a rectangle is calculated by multiplying its length and width. Given the area of a rectangular garden as $2x^2 + x - 6$ square feet, we need to find expressions that can represent the length and width of the garden. In this article, we will explore the relationship between the area, length, and width of a rectangular garden and determine which expressions can represent the length and width of the garden.

The Formula for the Area of a Rectangle

The area of a rectangle is given by the formula:

Area = Length × Width

We are given that the area of the rectangular garden is $2x^2 + x - 6$ square feet. Let's assume that the length of the garden is represented by the expression $L$ and the width is represented by the expression $W$. Then, we can write the equation:

$2x^2 + x - 6 = L \times W$

Finding the Length and Width of the Garden

To find the expressions that can represent the length and width of the garden, we need to factorize the given area expression. Let's factorize $2x^2 + x - 6$:

$2x^2 + x - 6 = (2x - 3)(x + 2)$

Now, we can see that the given area expression can be represented as the product of two expressions: $(2x - 3)$ and $(x + 2)$. These two expressions can represent the length and width of the garden.

Option A: Length = $x^2 - 3$ feet; Width = $2$ feet

Let's analyze the first option: Length = $x^2 - 3$ feet; Width = $2$ feet. We can see that the width is a constant value of $2$ feet, which is not a function of $x$. However, the length is represented by the expression $x^2 - 3$, which is a function of $x$. This option does not satisfy the condition that the length and width are both functions of $x$.

Option B: Length = $2x + 3$

Let's analyze the second option: Length = $2x + 3$. We can see that the length is represented by the expression $2x + 3$, which is a function of $x$. However, we need to find a corresponding width expression that is also a function of $x$. Let's assume that the width is represented by the expression $W$. Then, we can write the equation:

$2x^2 + x - 6 = (2x + 3) \times W$

We can see that the width expression $W$ must be a factor of the given area expression $2x^2 + x - 6$. Let's factorize $2x^2 + x - 6$:

$2x^2 + x - 6 = (2x - 3)(x + 2)$

We can see that the width expression $W$ can be represented by the expression $(2x - 3)$ or $(x + 2)$. Therefore, the length expression $2x + 3$ can be paired with either of these two width expressions.

Conclusion

In conclusion, the expressions that can represent the length and width of the garden are:

• Length = $2x + 3$; Width = $(2x - 3)$ or $(x + 2)$

These expressions satisfy the condition that the length and width are both functions of $x$. The other option, Length = $x^2 - 3$ feet; Width = $2$ feet, does not satisfy this condition.

Final Answer

The final answer is:

• Length = $2x + 3$; Width = $(2x - 3)$ or $(x + 2)$
  Frequently Asked Questions (FAQs) About the Length and Width of a Rectangular Garden =====================================================================================

Q: What is the relationship between the area, length, and width of a rectangular garden?

A: The area of a rectangle is calculated by multiplying its length and width. Given the area of a rectangular garden as $2x^2 + x - 6$ square feet, we need to find expressions that can represent the length and width of the garden.

Q: How do I find the expressions that can represent the length and width of the garden?

A: To find the expressions that can represent the length and width of the garden, we need to factorize the given area expression. Let's factorize $2x^2 + x - 6$:

$2x^2 + x - 6 = (2x - 3)(x + 2)$

Now, we can see that the given area expression can be represented as the product of two expressions: $(2x - 3)$ and $(x + 2)$. These two expressions can represent the length and width of the garden.

Q: What are the possible expressions that can represent the length and width of the garden?

A: The possible expressions that can represent the length and width of the garden are:

• Length = $2x + 3$; Width = $(2x - 3)$ or $(x + 2)$

These expressions satisfy the condition that the length and width are both functions of $x$.

Q: Why is it important to find the expressions that can represent the length and width of the garden?

A: Finding the expressions that can represent the length and width of the garden is important because it helps us understand the relationship between the area, length, and width of a rectangular garden. This knowledge can be applied to various real-world problems, such as designing gardens, calculating the cost of materials, and optimizing the use of space.

Q: Can I use other expressions to represent the length and width of the garden?

A: No, the expressions that can represent the length and width of the garden must be factors of the given area expression. In this case, the only possible expressions are:

• Length = $2x + 3$; Width = $(2x - 3)$ or $(x + 2)$

Q: How do I apply this knowledge to real-world problems?

A: You can apply this knowledge to real-world problems by using the expressions that can represent the length and width of the garden to calculate the area, perimeter, and other properties of the garden. For example, if you want to calculate the cost of materials for a garden, you can use the expressions for the length and width to calculate the area and then multiply it by the cost per square foot.

Q: What are some common mistakes to avoid when finding the expressions that can represent the length and width of the garden?

A: Some common mistakes to avoid when finding the expressions that can represent the length and width of the garden include:

• Not factorizing the given area expression
• Using expressions that are not factors of the given area expression
• Not considering the condition that the length and width are both functions of $x$

Conclusion

In conclusion, finding the expressions that can represent the length and width of a rectangular garden is an important concept in mathematics. By understanding the relationship between the area, length, and width of a rectangle, we can apply this knowledge to various real-world problems. Remember to factorize the given area expression, use expressions that are factors of the given area expression, and consider the condition that the length and width are both functions of $x$.