Solving For Missing Rectangle Side Length Using Quadratic Equations

In this article, we will walk through the process of solving a quadratic equation derived from a geometric problem. Specifically, we will address the problem where Clara writes the equation $(x-13)(x+8)=196$ to solve for the missing side length of a rectangle represented by the factor $x+8$. Our goal is to determine the missing side length represented by $x+8$ units of the rectangle. This involves understanding the relationship between the equation and the geometric context, manipulating the equation to find the value of $x$, and then substituting that value back into the expression $x+8$ to find the missing side length. Let's dive into the detailed steps to solve this problem effectively.

Understanding the Problem

Before we begin solving the equation, it's crucial to understand the context. The equation $(x-13)(x+8)=196$ represents the area of a rectangle, where $(x-13)$ and $(x+8)$ are the lengths of the sides, and 196 is the area. The unknown variable, $x$, is part of the expression $x+8$, which represents one of the side lengths we need to find. Understanding this context is key because it helps us interpret the mathematical solution in terms of a real-world geometric problem. When dealing with geometric problems translated into algebraic equations, it’s important to remember that negative lengths don’t make sense in the physical world. Therefore, any negative solutions we might find for $x$ will need to be carefully considered or discarded if they lead to a negative side length for the rectangle. This is a common consideration when solving quadratic equations that arise from geometric contexts, and it highlights the importance of always checking the reasonableness of your solutions within the original problem's framework. The connection between algebra and geometry here allows us to use algebraic techniques to solve for unknown dimensions, illustrating a fundamental principle in mathematical problem-solving. By recognizing that the quadratic equation models a rectangle's area, we can apply our knowledge of geometric properties to guide our algebraic manipulations and ensure that our final answer is not only mathematically correct but also physically meaningful in the context of the rectangle.

Expanding and Simplifying the Equation

The first step in solving for $x$ is to expand the left side of the equation $(x-13)(x+8)=196$. This involves using the distributive property (also known as the FOIL method) to multiply the two binomials. Specifically, we multiply each term in the first binomial by each term in the second binomial. This process is a fundamental algebraic technique used to simplify expressions and solve equations. The expansion will result in a quadratic expression, which we will then rearrange to set the equation equal to zero. This rearrangement is crucial because it sets the stage for using methods such as factoring, completing the square, or the quadratic formula to find the values of $x$. Expanding the equation correctly is paramount because any error in this step will propagate through the rest of the solution. Therefore, careful attention to detail is necessary to ensure that all terms are multiplied correctly and that signs are handled appropriately. The resulting quadratic equation will provide a clearer picture of the relationship between $x$ and the known area of the rectangle, allowing us to systematically solve for the unknown variable. This initial expansion is more than just a mechanical step; it's a critical transformation that brings the equation into a standard form that we can work with using well-established algebraic methods.

To expand $(x-13)(x+8)$, we follow these steps:

• Multiply the first terms: $x * x = x^2$
• Multiply the outer terms: $x * 8 = 8x$
• Multiply the inner terms: $-13 * x = -13x$
• Multiply the last terms: $-13 * 8 = -104$

Combining these results, we get $x^2 + 8x - 13x - 104 = 196$. Now, we simplify the equation by combining like terms: $x^2 - 5x - 104 = 196$. To solve a quadratic equation, we need to set it equal to zero. So, we subtract 196 from both sides of the equation: $x^2 - 5x - 104 - 196 = 0$. This simplifies to $x^2 - 5x - 300 = 0$.

Solving the Quadratic Equation

Now that we have the quadratic equation in the standard form $x^2 - 5x - 300 = 0$, 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, factoring is a viable option if we can find two numbers that multiply to -300 and add to -5. Alternatively, we can use the quadratic formula, which is a general method that works for any quadratic equation. Choosing the most efficient method can save time and effort, but understanding multiple methods provides flexibility in problem-solving. The quadratic formula is particularly useful when factoring is difficult or impossible. The choice of method also depends on the specific coefficients in the equation; some equations are more easily factored, while others require the quadratic formula or completing the square. Regardless of the method chosen, the goal is to find the values of $x$ that satisfy the equation. These values represent the potential solutions to our problem, but we must remember the context of the problem and discard any solutions that do not make sense in the context of the rectangle's side lengths. This step is critical in ensuring that our mathematical solution aligns with the real-world constraints of the geometric problem.

Let's solve the equation $x^2 - 5x - 300 = 0$ by factoring. We are looking for two numbers that multiply to -300 and add to -5. These numbers are -20 and 15. So, we can factor the quadratic equation as $(x - 20)(x + 15) = 0$. Setting each factor equal to zero gives us two possible solutions for $x$: $x - 20 = 0$ or $x + 15 = 0$. Solving these equations, we get $x = 20$ or $x = -15$.

Determining the Valid Solution

We have found two possible values for $x$: $x = 20$ and $x = -15$. However, we must consider the context of the problem. Since $x$ represents a length in a geometric figure, a negative value for $x$ does not make sense. Therefore, we discard the solution $x = -15$. This step is crucial because it highlights the importance of interpreting mathematical solutions within the context of the original problem. In many real-world applications, mathematical models involve physical quantities that cannot be negative, such as lengths, areas, or volumes. Discarding extraneous solutions is a key aspect of problem-solving in applied mathematics. It ensures that our final answer is not only mathematically correct but also physically meaningful. The process of checking the validity of solutions often involves going back to the original problem statement and considering the constraints and conditions given. This critical thinking step helps to refine the solution and avoid errors that might arise from blindly applying mathematical formulas. By rejecting the negative solution, we demonstrate a thorough understanding of the problem and a commitment to providing a practical and accurate answer.

Thus, the only valid solution is $x = 20$. Now, we need to find the missing side length represented by $x + 8$. Substituting $x = 20$ into the expression $x + 8$, we get $20 + 8 = 28$.

Final Answer: The Missing Side Length

After carefully solving the quadratic equation and considering the geometric context, we have determined that the missing side length represented by $x+8$ is 28 units. This final step involves substituting the valid value of $x$ back into the original expression to find the actual side length. It's a crucial step because it directly answers the question posed in the problem statement. The process of arriving at this answer demonstrates a comprehensive understanding of the problem-solving process, from translating the geometric problem into an algebraic equation to interpreting the mathematical solution in its original context. The final answer is not just a number; it's the solution to a real-world problem, and its units are important. In this case, the side length is measured in units, which could be inches, centimeters, or any other unit of length, depending on the specific context of the problem. Presenting the final answer clearly and including the appropriate units ensures that the solution is both accurate and easily understood. This concluding step reinforces the practical application of mathematical skills in solving geometric problems.

Therefore, the missing side length of the rectangle is 28 units.

Breakdown of the Solution

To recap, here’s a breakdown of the steps we took to solve this problem:

1. Understand the problem: We interpreted the equation $(x-13)(x+8) = 196$ as representing the area of a rectangle, where $x+8$ is one of the side lengths we need to find.
2. Expand and simplify the equation: We expanded the left side of the equation, combined like terms, and set the equation equal to zero, resulting in the quadratic equation $x^2 - 5x - 300 = 0$.
3. Solve the quadratic equation: We factored the quadratic equation to find the possible values of $x$: $x = 20$ and $x = -15$.
4. Determine the valid solution: We discarded the negative solution $x = -15$ because it doesn't make sense in the context of a side length.
5. Find the missing side length: We substituted the valid solution $x = 20$ into the expression $x + 8$ to find the missing side length, which is 28 units.

This step-by-step approach demonstrates a systematic method for solving mathematical problems that involve translating real-world scenarios into algebraic equations. Each step is crucial in ensuring the accuracy and validity of the final solution. Breaking down the solution into smaller, manageable steps makes the problem-solving process more approachable and less daunting. It also allows for easier identification of potential errors and facilitates a deeper understanding of the underlying mathematical concepts. The ability to break down complex problems into simpler steps is a valuable skill in mathematics and many other fields, as it promotes clarity, organization, and efficiency in problem-solving.

Answer

The missing side length represented by $x+8$ units of the rectangle is 28.

Keywords

Rectangle side length, quadratic equation, solving for x, factoring, geometric problem, missing side, positive solution, extraneous solution