Solving For Rectangle Dimensions Using Quadratic Equations (x+5)x = 104
In the realm of geometry, rectangles stand as fundamental shapes, their properties and dimensions captivating mathematicians and enthusiasts alike. This article delves into a specific problem involving a rectangle whose length surpasses its width by 5 inches, with an area of 104 square inches. The scenario is elegantly captured by the equation (x+5)x = 104, where x represents the elusive width of the rectangle. Our exploration will unravel the steps to solve this equation, revealing the rectangle's dimensions and providing a deeper understanding of quadratic equations in practical applications. We'll embark on a step-by-step journey, transforming the equation into a standard quadratic form, employing the quadratic formula to extract the roots, and finally, interpreting these roots within the geometric context to determine the rectangle's width and length. This endeavor not only sharpens our algebraic skills but also illuminates the seamless connection between mathematical models and real-world geometric scenarios.
Setting up the Equation: Translating Words into Math
The journey to solve any mathematical problem begins with translating the given information into a symbolic representation. In our case, we are presented with a rectangle, a geometric figure characterized by its length and width. A crucial piece of information is that the length exceeds the width by 5 inches. This relationship can be elegantly expressed using a variable: let x represent the width of the rectangle. Consequently, the length can be represented as x + 5. The area of a rectangle, a measure of the space it occupies, is calculated by multiplying its length and width. We are given that the area is 104 square inches. Thus, we arrive at the equation (x + 5)x = 104. This equation is the cornerstone of our problem, encapsulating the relationship between the rectangle's dimensions and its area. The left-hand side, (x + 5)x, represents the product of the length and width, while the right-hand side, 104, represents the given area. This equation is a quadratic equation, a type of equation that involves a variable raised to the power of 2. Quadratic equations are ubiquitous in mathematics and have a wide range of applications, from physics to engineering to economics. Solving this equation will reveal the value of x, which in turn will allow us to determine the width and length of the rectangle. This initial step of setting up the equation is paramount, as it lays the foundation for the subsequent algebraic manipulations that will lead us to the solution.
Transforming to Standard Form: Preparing for the Quadratic Formula
Before we can unleash the power of the quadratic formula, our equation needs to be in a specific format: the standard quadratic form. This form is expressed as ax² + bx + c = 0, where a, b, and c are constants. Our equation, (x + 5)x = 104, is not in this form yet. To transform it, we need to perform some algebraic maneuvers. First, we distribute the x on the left-hand side, multiplying it with both terms inside the parentheses: x * x + 5 * x = 104. This simplifies to x² + 5x = 104. Now, to achieve the standard form, we need to bring all terms to one side of the equation, leaving zero on the other side. We accomplish this by subtracting 104 from both sides: x² + 5x - 104 = 104 - 104. This results in the equation x² + 5x - 104 = 0. Behold, our equation is now in the coveted standard quadratic form! We can clearly identify the coefficients: a = 1, b = 5, and c = -104. These coefficients are the keys that will unlock the solutions using the quadratic formula. The transformation to standard form is a crucial step because the quadratic formula is specifically designed to solve equations in this format. It's like preparing the ingredients before baking a cake; we need the equation in the right form before we can apply the solving technique.
Unleashing the Quadratic Formula: Finding the Roots
With our equation nestled snugly in the standard quadratic form, we are now ready to wield the mighty quadratic formula. This formula is a mathematical marvel, a universal tool for solving quadratic equations of the form ax² + bx + c = 0. It states that the solutions for x are given by: x = (-b ± √(b² - 4ac)) / 2a. Let's break down this formula and see how it applies to our equation, x² + 5x - 104 = 0. We've already identified our coefficients: a = 1, b = 5, and c = -104. Now, we carefully substitute these values into the quadratic formula: x = (-5 ± √(5² - 4 * 1 * -104)) / (2 * 1). Next, we simplify the expression under the square root: 5² - 4 * 1 * -104 = 25 + 416 = 441. So, our equation becomes: x = (-5 ± √441) / 2. The square root of 441 is 21, so we have: x = (-5 ± 21) / 2. Now, the ± symbol indicates that we have two possible solutions, one with a plus sign and one with a minus sign. Let's calculate both: x₁ = (-5 + 21) / 2 = 16 / 2 = 8 and x₂ = (-5 - 21) / 2 = -26 / 2 = -13. Thus, we have found two roots, x₁ = 8 and x₂ = -13. The quadratic formula has delivered the solutions, but our journey isn't over yet. We need to interpret these roots within the context of our geometric problem to determine the actual dimensions of the rectangle. The quadratic formula is a powerful tool, but it's important to remember that the solutions it provides are mathematical abstractions. We need to connect these abstractions back to the real-world scenario we are trying to model.
Interpreting the Roots: Finding the Rectangle's Dimensions
We've arrived at two potential solutions for x: 8 and -13. But what do these numbers mean in the context of our rectangle? Remember, x represents the width of the rectangle. Can a rectangle have a negative width? In the realm of real-world geometry, the answer is a resounding no. Dimensions like length and width must be positive values. Therefore, we discard the negative root, x = -13, as it doesn't make sense in our physical context. This leaves us with x = 8 as the valid solution for the width. Now that we know the width, we can easily find the length. The length is given by x + 5, so the length is 8 + 5 = 13 inches. We have successfully unveiled the dimensions of our rectangle! The width is 8 inches, and the length is 13 inches. But let's not stop here. It's always a good practice to verify our solution. We can do this by calculating the area of the rectangle using the dimensions we found: Area = length * width = 13 inches * 8 inches = 104 square inches. This matches the given area in the problem, confirming that our solution is indeed correct. Interpreting the roots within the context of the problem is a crucial step in mathematical problem-solving. It's not enough to just find the solutions; we need to understand what they mean and whether they make sense in the real-world situation we are modeling.
Conclusion: A Geometric Puzzle Solved
In this exploration, we embarked on a journey to unravel the dimensions of a rectangle, guided by the equation (x+5)x = 104. We successfully navigated the algebraic landscape, transforming the equation into standard quadratic form, employing the quadratic formula to extract the roots, and carefully interpreting these roots within the geometric context. We discovered that the width of the rectangle is 8 inches and the length is 13 inches. This problem showcased the power of mathematical modeling, where equations can capture real-world scenarios and provide solutions. It also highlighted the importance of careful interpretation, ensuring that our mathematical results align with the physical constraints of the problem. Quadratic equations, like the one we solved, are fundamental tools in mathematics and have a wide range of applications in various fields. Mastering the techniques to solve them opens doors to understanding and solving a multitude of problems in science, engineering, and beyond. The journey from setting up the equation to interpreting the roots is a testament to the problem-solving process, a process that involves translating words into symbols, manipulating equations, and finally, connecting the abstract world of mathematics to the concrete world around us. This is the essence of mathematical thinking, a skill that is valuable not only in the classroom but also in everyday life.
