A Rectangle Has An Area Given By $x^2 - 4x - 12$. If The Length And Width Of The Rectangle Are Each Increased By 3, What Is The Area Of The New, Larger Rectangle In Simplified Form?A. $x^2 - 4x - 6$ B. $3x^2 - 12x -
Introduction
When dealing with geometric shapes, understanding the relationships between their dimensions and properties is crucial. In this article, we will delve into the world of rectangles and explore how changes in their dimensions affect their area. Specifically, we will examine the scenario where the length and width of a rectangle are each increased by 3, and determine the area of the new, larger rectangle in simplified form.
The Original Rectangle
The original rectangle has an area given by the quadratic expression $x^2 - 4x - 12$. This expression represents the product of the length and width of the rectangle. To find the area of the new, larger rectangle, we need to consider the impact of increasing the length and width by 3.
Increasing the Dimensions
When the length and width of the rectangle are each increased by 3, the new dimensions can be represented as $(x+3)$ and $(x+3)$, respectively. To find the area of the new rectangle, we need to multiply these new dimensions together.
Calculating the New Area
The area of the new rectangle can be calculated by multiplying the new length and width:
Expanding the squared expression, we get:
Simplifying the Expression
To simplify the expression, we can combine like terms:
However, we can further simplify the expression by factoring out the common term $x^2$:
The Final Answer
The area of the new, larger rectangle in simplified form is:
However, this is not among the answer choices provided. Let's re-examine the original expression and see if we can find a connection between it and the new area.
Revisiting the Original Expression
The original expression is $x^2 - 4x - 12$. We can try to factor this expression to see if it reveals any connections to the new area:
Finding the Connection
Now that we have factored the original expression, we can see that it can be written as the product of two binomials:
This suggests that the original rectangle can be thought of as having a length of $(x-6)$ and a width of $(x+2)$.
Increasing the Dimensions Again
When we increase the length and width by 3, we get:
Calculating the New Area Again
The area of the new rectangle can be calculated by multiplying the new length and width:
The Final Answer Revisited
The area of the new, larger rectangle in simplified form is:
This is not among the answer choices provided, but it is a valid solution.
Conclusion
In this article, we explored the impact of increasing the dimensions of a rectangle on its area. We started with a quadratic expression representing the original area and then increased the length and width by 3. We calculated the new area and simplified the expression to find the final answer. Our journey revealed the importance of factoring and understanding the relationships between the dimensions and properties of geometric shapes.
Answer
The correct answer is not among the options provided. However, based on our calculations, the area of the new, larger rectangle in simplified form is:
This is a valid solution, but it is not among the answer choices provided.
Introduction
In our previous article, we explored the impact of increasing the dimensions of a rectangle on its area. We started with a quadratic expression representing the original area and then increased the length and width by 3. We calculated the new area and simplified the expression to find the final answer. In this article, we will answer some frequently asked questions related to the topic.
Q: What is the original area of the rectangle?
A: The original area of the rectangle is given by the quadratic expression $x^2 - 4x - 12$.
Q: How do you increase the dimensions of a rectangle?
A: To increase the dimensions of a rectangle, you need to add a fixed value to both the length and width. In this case, we increased the length and width by 3.
Q: How do you calculate the new area of a rectangle?
A: To calculate the new area of a rectangle, you need to multiply the new length and width together.
Q: What is the new area of the rectangle in simplified form?
A: The new area of the rectangle in simplified form is $x^2 + 2x - 15$.
Q: Why is the new area not among the answer choices provided?
A: The new area is not among the answer choices provided because the question was not properly framed. The correct answer is $x^2 + 2x - 15$, but it was not among the options.
Q: What is the importance of factoring in this problem?
A: Factoring is important in this problem because it helps us understand the relationships between the dimensions and properties of geometric shapes. By factoring the original expression, we were able to reveal connections between the original area and the new area.
Q: Can you provide more examples of increasing the dimensions of a rectangle?
A: Yes, we can provide more examples. For instance, if we increase the length and width by 2, the new area would be $x^2 + 4x - 8$. If we increase the length and width by 4, the new area would be $x^2 + 8x - 16$.
Q: How do you determine the correct answer in a problem like this?
A: To determine the correct answer in a problem like this, you need to carefully read the question and understand what is being asked. You also need to check your work and make sure that your answer is correct.
Q: What are some common mistakes to avoid when increasing the dimensions of a rectangle?
A: Some common mistakes to avoid when increasing the dimensions of a rectangle include:
- Not increasing both the length and width by the same amount
- Not multiplying the new length and width together to get the new area
- Not checking your work to make sure that your answer is correct
Conclusion
In this article, we answered some frequently asked questions related to the topic of increasing the dimensions of a rectangle. We provided examples and explanations to help clarify the concepts and avoid common mistakes. By following these tips and understanding the relationships between the dimensions and properties of geometric shapes, you can become more confident and proficient in solving problems like this.
Additional Resources
If you want to learn more about increasing the dimensions of a rectangle, we recommend checking out the following resources:
Final Answer
The final answer is $x^2 + 2x - 15$.