Area Of Rectangle = 5x²-9-12 Breadth = ?Length = X-4
Introduction
In mathematics, the area of a rectangle is a fundamental concept that is used to calculate the size of a rectangle. The area of a rectangle is given by the formula: Area = Length × Breadth. In this article, we will explore how to solve for the breadth of a rectangle when the area and length are given. We will use the given equation: Area = 5x² - 9 - 12, and Length = x - 4.
Understanding the Given Equation
The given equation is: Area = 5x² - 9 - 12. This equation represents the area of the rectangle in terms of the variable x. To solve for the breadth, we need to isolate the breadth term in the equation. However, the equation is not in the standard form of a quadratic equation, which is ax² + bx + c = 0. We need to simplify the equation to make it easier to solve.
Simplifying the Equation
To simplify the equation, we can combine the constant terms: -9 - 12 = -21. The simplified equation is: Area = 5x² - 21.
Substituting the Length
We are given that the length of the rectangle is x - 4. We can substitute this expression into the equation: Area = Length × Breadth. Substituting the length, we get: 5x² - 21 = (x - 4) × Breadth.
Expanding the Equation
To expand the equation, we can multiply the length by the breadth: (x - 4) × Breadth = x × Breadth - 4 × Breadth. The expanded equation is: 5x² - 21 = x × Breadth - 4 × Breadth.
Simplifying the Equation
To simplify the equation, we can combine the breadth terms: x × Breadth - 4 × Breadth = (x - 4) × Breadth. The simplified equation is: 5x² - 21 = (x - 4) × Breadth.
Solving for the Breadth
To solve for the breadth, we need to isolate the breadth term in the equation. We can do this by dividing both sides of the equation by (x - 4): Breadth = (5x² - 21) / (x - 4).
Simplifying the Equation
To simplify the equation, we can factor the numerator: 5x² - 21 = 5(x² - 4.2). The simplified equation is: Breadth = 5(x² - 4.2) / (x - 4).
Solving for the Breadth
To solve for the breadth, we need to simplify the equation further. We can do this by factoring the numerator: x² - 4.2 = (x - √4.2)(x + √4.2). The simplified equation is: Breadth = 5(x - √4.2)(x + √4.2) / (x - 4).
Simplifying the Equation
To simplify the equation, we can cancel out the common factor (x - 4): Breadth = 5(x + √4.2) / 1.
Solving for the Breadth
To solve for the breadth, we need to simplify the equation further. We can do this by multiplying the numerator and denominator by 1: Breadth = 5(x + √4.2).
Simplifying the Equation
To simplify the equation, we can multiply the numerator and denominator by 1: Breadth = 5(x + √4.2).
The Final Answer
The final answer is: Breadth = 5(x + √4.2).
Conclusion
In this article, we explored how to solve for the breadth of a rectangle when the area and length are given. We used the given equation: Area = 5x² - 9 - 12, and Length = x - 4. We simplified the equation, substituted the length, expanded the equation, simplified the equation, and solved for the breadth. The final answer is: Breadth = 5(x + √4.2).
Real-World Applications
The concept of solving for the breadth of a rectangle has many real-world applications. For example, in architecture, engineers need to calculate the size of a building's walls to ensure that they are strong enough to support the weight of the building. In manufacturing, companies need to calculate the size of their products to ensure that they fit together properly. In construction, builders need to calculate the size of the materials they need to build a structure.
Tips and Tricks
When solving for the breadth of a rectangle, it is essential to simplify the equation as much as possible. This can be done by combining like terms, factoring the numerator, and canceling out common factors. It is also essential to check the solution to ensure that it is reasonable and makes sense in the context of the problem.
Common Mistakes
When solving for the breadth of a rectangle, there are several common mistakes that can be made. One of the most common mistakes is to forget to simplify the equation. Another common mistake is to forget to check the solution to ensure that it is reasonable and makes sense in the context of the problem.
Conclusion
In conclusion, solving for the breadth of a rectangle is a fundamental concept in mathematics that has many real-world applications. By simplifying the equation, substituting the length, expanding the equation, simplifying the equation, and solving for the breadth, we can find the final answer. The final answer is: Breadth = 5(x + √4.2).
Q: What is the formula for the area of a rectangle?
A: The formula for the area of a rectangle is: Area = Length × Breadth.
Q: How do I solve for the breadth of a rectangle when the area and length are given?
A: To solve for the breadth of a rectangle when the area and length are given, you need to use the formula: Area = Length × Breadth. You can then substitute the given values into the equation and solve for the breadth.
Q: What is the given equation in this problem?
A: The given equation in this problem is: Area = 5x² - 9 - 12, and Length = x - 4.
Q: How do I simplify the equation?
A: To simplify the equation, you can combine like terms, factor the numerator, and cancel out common factors.
Q: What is the final answer for the breadth of the rectangle?
A: The final answer for the breadth of the rectangle is: Breadth = 5(x + √4.2).
Q: What are some real-world applications of solving for the breadth of a rectangle?
A: Some real-world applications of solving for the breadth of a rectangle include architecture, manufacturing, and construction.
Q: What are some common mistakes to avoid when solving for the breadth of a rectangle?
A: Some common mistakes to avoid when solving for the breadth of a rectangle include forgetting to simplify the equation, forgetting to check the solution, and not using the correct formula.
Q: How do I check the solution to ensure that it is reasonable and makes sense in the context of the problem?
A: To check the solution, you can plug the values back into the original equation and ensure that the solution satisfies the equation.
Q: What is the importance of simplifying the equation when solving for the breadth of a rectangle?
A: Simplifying the equation is essential when solving for the breadth of a rectangle because it makes it easier to solve for the breadth and ensures that the solution is reasonable and makes sense in the context of the problem.
Q: Can I use a calculator to solve for the breadth of a rectangle?
A: Yes, you can use a calculator to solve for the breadth of a rectangle. However, it is essential to check the solution to ensure that it is reasonable and makes sense in the context of the problem.
Q: What is the difference between the breadth and the length of a rectangle?
A: The breadth and the length of a rectangle are two different dimensions of the rectangle. The breadth is the width of the rectangle, while the length is the distance between the two opposite sides of the rectangle.
Q: Can I use the same formula to solve for the length of a rectangle?
A: Yes, you can use the same formula to solve for the length of a rectangle. However, you will need to substitute the given values into the equation and solve for the length.
Q: What are some other formulas that I can use to solve for the dimensions of a rectangle?
A: Some other formulas that you can use to solve for the dimensions of a rectangle include the formula for the perimeter of a rectangle (P = 2L + 2B) and the formula for the area of a rectangle (A = L × B).
Q: Can I use the same method to solve for the dimensions of other shapes, such as triangles and circles?
A: No, the same method cannot be used to solve for the dimensions of other shapes, such as triangles and circles. Each shape has its own unique formulas and methods for solving for its dimensions.
Q: What are some other mathematical concepts that I can use to solve for the dimensions of a rectangle?
A: Some other mathematical concepts that you can use to solve for the dimensions of a rectangle include algebra, geometry, and trigonometry.
Q: Can I use a graphing calculator to solve for the dimensions of a rectangle?
A: Yes, you can use a graphing calculator to solve for the dimensions of a rectangle. However, it is essential to check the solution to ensure that it is reasonable and makes sense in the context of the problem.
Q: What are some other real-world applications of solving for the dimensions of a rectangle?
A: Some other real-world applications of solving for the dimensions of a rectangle include engineering, physics, and computer science.