Solving For Missing Side Length Clara's Rectangle Problem

Introduction

In this article, we will delve into a mathematical problem where Clara uses an equation to determine the missing side length of a rectangle. The equation she employs is $(x-13)(x+8)=196$, and our goal is to find the value of the side length represented by the factor $x+8$. This problem combines algebraic manipulation with geometric concepts, offering a practical application of quadratic equations. Understanding how to solve such problems is crucial for various fields, including engineering, architecture, and even everyday problem-solving. By breaking down the equation and systematically solving for $x$, we can then determine the missing side length of the rectangle. This exploration will not only enhance our mathematical skills but also provide insights into how algebraic equations can model real-world geometric scenarios. The following sections will guide you through the step-by-step process of solving this equation and interpreting the result in the context of the rectangle's dimensions. We will begin by expanding the equation, simplifying it, and then using algebraic techniques to find the value of $x$. Once we have the value of $x$, we can easily calculate the missing side length represented by $x+8$. This article aims to provide a clear and comprehensive explanation, making it accessible for anyone interested in mathematics and problem-solving.

Understanding the Problem

To begin, let's break down the problem statement. Clara is using the equation $(x-13)(x+8)=196$ to find a missing side length of a rectangle. The expression $x+8$ represents one of the rectangle's sides, and our objective is to determine the length of this side. The equation itself suggests that the product of two expressions, $(x-13)$ and $(x+8)$, equals 196. This setup is indicative of a quadratic equation, which we will need to solve to find the value of $x$. Understanding the context of the problem is crucial. We are dealing with a rectangle, a geometric shape with specific properties, such as having opposite sides of equal length and four right angles. The dimensions of the rectangle are related through the variable $x$, and solving for $x$ will give us the numerical values of these dimensions. This problem highlights the intersection of algebra and geometry, where algebraic equations are used to model geometric relationships. Before diving into the algebraic manipulations, it's important to visualize the rectangle and understand what each part of the equation represents in terms of the rectangle's sides. This visual and conceptual understanding will make the solution process more intuitive and meaningful. The next step involves expanding and simplifying the equation to a standard quadratic form, which will allow us to use various methods to solve for $x$.

Solving the Quadratic Equation

The equation $(x-13)(x+8)=196$ is the key to finding the missing side length. To solve for $x$, we first need to expand the equation. Expanding the left side, we get: $(x-13)(x+8) = x^2 + 8x - 13x - 104$. Combining like terms, this simplifies to $x^2 - 5x - 104$. Now, we set the entire equation equal to zero by subtracting 196 from both sides: $x^2 - 5x - 104 - 196 = 0$. This further simplifies to $x^2 - 5x - 300 = 0$. We now have a standard quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = -5$, and $c = -300$. To solve this quadratic equation, we can use factoring, the quadratic formula, or completing the square. Factoring is often the quickest method if we can find two numbers that multiply to $c$ and add to $b$. In this case, we are looking for two numbers that multiply to -300 and add to -5. Those numbers are -20 and 15. Therefore, we can factor the quadratic equation as follows: $(x - 20)(x + 15) = 0$. Setting each factor equal to zero gives us two possible solutions for $x$: $x - 20 = 0$ which gives $x = 20$, and $x + 15 = 0$ which gives $x = -15$. These are the two possible values for $x$ that satisfy the equation.

Determining the Valid Solution

We have found two possible values for $x$: $x = 20$ and $x = -15$. However, in the context of this problem, we need to consider which solution makes sense. Since $x$ represents a length, it cannot be negative. Therefore, $x = -15$ is not a valid solution for this problem. This leaves us with $x = 20$ as the only viable solution. Now that we have the value of $x$, we can determine the missing side length represented by $x + 8$. Substituting $x = 20$ into the expression $x + 8$, we get $20 + 8 = 28$. Therefore, the missing side length of the rectangle is 28 units. It is crucial to consider the context of the problem when interpreting solutions. In geometric problems, lengths and distances cannot be negative, so we must discard any negative solutions. This step of verifying the solution ensures that our answer is not only mathematically correct but also physically meaningful. Understanding the constraints of the problem helps us to filter out extraneous solutions and arrive at the correct answer. In this case, the valid solution $x = 20$ allows us to find the missing side length, which is a positive value, as expected for a geometric dimension. This process of elimination and validation is a critical aspect of mathematical problem-solving.

Calculating the Missing Side Length

Having determined that $x = 20$, we can now calculate the missing side length represented by $x + 8$. Substituting $x = 20$ into the expression, we get: $x + 8 = 20 + 8 = 28$. Therefore, the missing side length of the rectangle is 28 units. This result provides the numerical answer to the problem. To further understand the rectangle, we can also find the other side length using the expression $x - 13$. Substituting $x = 20$, we get: $x - 13 = 20 - 13 = 7$. So, the other side length of the rectangle is 7 units. Now we know both side lengths of the rectangle are 7 units and 28 units. We can verify our solution by checking if the product of the original expressions $(x - 13)$ and $(x + 8)$ equals 196 when $x = 20$: $(20 - 13)(20 + 8) = (7)(28) = 196$. This confirms that our solution is correct. The missing side length, represented by $x + 8$, is indeed 28 units. This step-by-step calculation and verification process ensures the accuracy of our answer and demonstrates a thorough understanding of the problem. The ability to substitute values and check results is an essential skill in mathematics, reinforcing the validity of our solutions.

Conclusion

In conclusion, we successfully solved the problem presented by Clara. By expanding the equation $(x-13)(x+8)=196$, simplifying it to a quadratic form, and solving for $x$, we found two possible solutions: $x = 20$ and $x = -15$. However, considering the context of the problem, we discarded the negative solution because side lengths cannot be negative. The valid solution, $x = 20$, allowed us to calculate the missing side length represented by $x + 8$, which is 28 units. This problem demonstrates the practical application of quadratic equations in geometric contexts. It highlights the importance of not only solving equations correctly but also interpreting the solutions within the given context. The ability to translate algebraic expressions into geometric dimensions and vice versa is a valuable skill in mathematics and its applications. By breaking down the problem into smaller, manageable steps, we were able to systematically arrive at the correct answer. This process reinforces the importance of a logical and methodical approach to problem-solving. The problem also illustrates the significance of verifying solutions to ensure they are both mathematically sound and physically meaningful. Through this exploration, we have enhanced our understanding of quadratic equations and their relevance in real-world scenarios, particularly in geometric problems involving rectangles and their dimensions.