Davidson Family Patio Expansion Solving Quadratic Equations

Introduction

The Davidson family is embarking on a home improvement project, aiming to enhance their outdoor living space by expanding their rectangular patio. Currently, the patio measures 15 feet in length and 12 feet in width. The family envisions extending both the length and width by the same amount, seeking to increase the patio's total area by a significant 160 square feet. This scenario presents an interesting mathematical challenge that can be effectively addressed using a quadratic equation. In this comprehensive article, we will delve into the step-by-step process of formulating and solving the quadratic equation that models this patio expansion problem. Our exploration will cover the initial setup, the derivation of the equation, the solution using various methods, and a practical interpretation of the results. By the end of this article, you will gain a solid understanding of how quadratic equations can be applied to solve real-world problems, particularly those involving geometric expansions and area calculations. This problem not only demonstrates the practical utility of quadratic equations but also highlights the importance of mathematical modeling in everyday situations. Whether you are a student learning about quadratic equations or a homeowner planning a similar expansion project, this article provides valuable insights and a clear methodology for tackling such challenges. Let's begin by carefully outlining the problem and defining the variables that will help us translate the scenario into a mathematical framework.

Problem Statement and Initial Setup

To start, let's clearly define the problem. The Davidson family's existing patio is a rectangle with dimensions 15 feet by 12 feet. The area of this patio is simply the product of its length and width, which is 15 ft * 12 ft = 180 sq ft. The family wants to increase this area by 160 sq ft, so the target area for the expanded patio is 180 sq ft + 160 sq ft = 340 sq ft. The key to solving this problem lies in understanding how the dimensions of the patio change when both the length and width are extended by the same amount. Let's denote the amount by which the length and width are extended as 'x' feet. This 'x' is our variable, and it represents the unknown quantity we need to find. When the length is extended by x feet, the new length becomes (15 + x) feet. Similarly, when the width is extended by x feet, the new width becomes (12 + x) feet. The area of the expanded patio can then be expressed as the product of the new length and the new width, which is (15 + x)(12 + x). Now, we know that this new area must equal 340 sq ft. This gives us the foundation for our quadratic equation. Setting up the equation correctly is crucial for solving the problem accurately. The equation will relate the expanded area (15 + x)(12 + x) to the target area of 340 sq ft. This equation will be a quadratic equation because when we expand the product (15 + x)(12 + x), we will get a term with x squared. The next step is to actually form this equation and put it in the standard quadratic form, which is ax^2 + bx + c = 0. This standard form is essential because it allows us to use various methods, such as factoring, completing the square, or the quadratic formula, to find the solutions for x. Understanding the problem thoroughly and setting up the equation correctly are the most critical steps in solving any mathematical problem. With the equation in hand, we can then proceed to solve for x and find the amount by which the patio needs to be extended.

Deriving the Quadratic Equation

Now, let's derive the quadratic equation that models the Davidson family's patio expansion. As established earlier, the area of the expanded patio is given by the product of its new length and new width, which are (15 + x) feet and (12 + x) feet, respectively. The total area of the expanded patio is the sum of the original area (180 sq ft) and the additional area (160 sq ft), which totals 340 sq ft. Therefore, we can write the equation as: (15 + x)(12 + x) = 340. To transform this equation into the standard quadratic form (ax^2 + bx + c = 0), we need to expand the product on the left side and then rearrange the terms. First, we expand (15 + x)(12 + x) using the distributive property (also known as the FOIL method): (15 + x)(12 + x) = 15 * 12 + 15 * x + x * 12 + x * x = 180 + 15x + 12x + x^2. Combining the like terms, we get: x^2 + 27x + 180. Now, we set this expression equal to the target area, 340 sq ft: x^2 + 27x + 180 = 340. To get the equation into the standard quadratic form, we need to subtract 340 from both sides: x^2 + 27x + 180 - 340 = 0. This simplifies to: x^2 + 27x - 160 = 0. This is the quadratic equation that we need to solve. Here, 'a' is 1, 'b' is 27, and 'c' is -160. This equation represents the relationship between the amount of extension 'x' and the desired increase in area. Solving this equation will give us the value(s) of 'x' that satisfy the condition that the patio's area increases by 160 sq ft. With the equation in the standard form, we can now choose an appropriate method to solve for 'x'. The common methods include factoring, completing the square, and using the quadratic formula. Each method has its advantages and disadvantages, and the choice of method often depends on the specific coefficients of the quadratic equation. In the next section, we will explore these methods and apply one of them to find the solution for 'x'.

Solving the Quadratic Equation

Now that we have the quadratic equation x^2 + 27x - 160 = 0, we need to solve for x. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, the quadratic formula is the most straightforward method due to the coefficients of the equation. The quadratic formula is given by: x = [-b ± √(b^2 - 4ac)] / (2a). Where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our equation, a = 1, b = 27, and c = -160. Plugging these values into the quadratic formula, we get: x = [-27 ± √(27^2 - 4 * 1 * (-160))] / (2 * 1). First, let's calculate the discriminant (the expression inside the square root): 27^2 - 4 * 1 * (-160) = 729 + 640 = 1369. Now, we can find the square root of 1369, which is 37. So, the equation becomes: x = [-27 ± 37] / 2. This gives us two possible solutions for x: x1 = (-27 + 37) / 2 = 10 / 2 = 5 and x2 = (-27 - 37) / 2 = -64 / 2 = -32. We have two values for x: 5 and -32. However, in the context of this problem, x represents the amount by which the length and width of the patio are extended. Since we cannot have a negative extension, we discard the negative solution (-32). Therefore, the only valid solution is x = 5. This means that the Davidson family needs to extend both the length and the width of their patio by 5 feet to increase the area by 160 sq ft. We can verify this solution by calculating the dimensions of the expanded patio and its area. The new length will be 15 + 5 = 20 feet, and the new width will be 12 + 5 = 17 feet. The area of the expanded patio is 20 ft * 17 ft = 340 sq ft, which is exactly 160 sq ft more than the original area of 180 sq ft. Thus, our solution is correct. In the next section, we will discuss the practical implications of this solution and how it addresses the Davidson family's patio expansion project.

Practical Implications and Conclusion

Having solved the quadratic equation, we found that x = 5 feet. This means the Davidson family needs to extend both the length and width of their patio by 5 feet to achieve their goal of increasing the patio area by 160 sq ft. Let's consider the practical implications of this solution. The original dimensions of the patio were 15 feet by 12 feet. By extending each side by 5 feet, the new dimensions become: New length = 15 feet + 5 feet = 20 feet. New width = 12 feet + 5 feet = 17 feet. The new area of the patio is therefore 20 feet * 17 feet = 340 sq ft. As we calculated earlier, this is an increase of 160 sq ft from the original area of 180 sq ft (15 feet * 12 feet). This solution provides a clear plan for the Davidson family. They know precisely how much they need to extend their patio to meet their desired area increase. This is a practical and actionable outcome derived directly from the mathematical solution. Furthermore, this example illustrates the power of using mathematical models, specifically quadratic equations, to solve real-world problems. The ability to translate a practical problem into a mathematical equation allows us to use the tools of algebra to find precise solutions. In this case, the quadratic equation provided a straightforward method to determine the required extension for the patio. In conclusion, the Davidson family's patio expansion problem is a great example of how quadratic equations can be applied in everyday situations. By setting up and solving the equation x^2 + 27x - 160 = 0, we determined that extending the patio by 5 feet in both length and width would increase the area by 160 sq ft. This solution not only answers the specific question posed but also demonstrates the broader applicability of mathematical concepts in practical scenarios. Whether you're planning a home improvement project or solving a theoretical problem, understanding how to use mathematical tools like quadratic equations can be incredibly valuable. This exercise highlights the importance of mathematical literacy and problem-solving skills in various aspects of life.