Factoring Quadratic Expressions Finding Rectangle Dimensions

In this comprehensive exploration, we delve into the practical application of factoring quadratic expressions by examining a geometric problem. Imagine Logan, who, after some initial missteps, is now prepared to tackle factoring. He faces the challenge of determining the dimensions of a rectangle given its area, expressed as the quadratic expression $3x^2 - 13x - 10$. Our task is to guide Logan through the factoring process, ultimately revealing the length and width of the rectangle. This problem not only reinforces factoring skills but also illustrates how algebraic concepts translate into real-world geometric scenarios.

Understanding the Problem Area and Dimensions

The initial step in solving this problem is to fully grasp the connection between the area of a rectangle and its dimensions. The area of a rectangle is calculated by multiplying its length and width. In algebraic terms, if we represent the length as L and the width as W, the area A is given by the formula: $A = L \times W$. In Logan's case, the area A is provided as a quadratic expression, $3x^2 - 13x - 10$. To find the dimensions, we need to factor this quadratic expression into two binomials, which will represent the length and width of the rectangle. This process involves reversing the multiplication process, breaking down the quadratic expression into its constituent factors. By understanding this fundamental relationship, Logan can approach the problem with a clear strategy, recognizing that factoring is the key to unlocking the rectangle's dimensions. The ability to translate a geometric problem into an algebraic equation and vice versa is a crucial skill in mathematics, bridging the gap between abstract concepts and concrete applications. This understanding sets the stage for the subsequent steps in the factoring process, ensuring that Logan can confidently navigate the challenges ahead.

Factoring Quadratic Expressions A Step-by-Step Guide

To successfully factor the quadratic expression $3x^2 - 13x - 10$, Logan needs to employ a systematic approach. The most common method for factoring such expressions involves several key steps, which we will outline in detail. Firstly, identify the coefficients a, b, and c in the quadratic expression $ax^2 + bx + c$. In this case, a = 3, b = -13, and c = -10. Next, calculate the product of a and c, which is $3 \times -10 = -30$. The goal is to find two numbers that multiply to this product (-30) and add up to b (-13). After some consideration, we identify the numbers -15 and 2, as $(-15) \times 2 = -30$ and $-15 + 2 = -13$. Now, rewrite the middle term (-13x) using these two numbers: $3x^2 - 15x + 2x - 10$. This step is crucial as it allows us to group terms and factor by grouping. Group the first two terms and the last two terms: $(3x^2 - 15x) + (2x - 10)$. Factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 3x, and from the second group, the GCF is 2. This gives us $3x(x - 5) + 2(x - 5)$. Notice that both terms now have a common factor of $(x - 5)$. Factor out this common binomial factor: $(x - 5)(3x + 2)$. This final expression represents the factored form of the original quadratic expression. The binomials $(x - 5)$ and $(3x + 2)$ represent the potential dimensions of the rectangle. By carefully following these steps, Logan can confidently factor quadratic expressions and apply this skill to various problems.

Applying Factoring to Find Dimensions The Solution

Now that we have factored the quadratic expression $3x^2 - 13x - 10$ into $(x - 5)(3x + 2)$, we can determine the dimensions of the rectangle. The factors $(x - 5)$ and $(3x + 2)$ represent the length and width of the rectangle. It is important to note that in real-world contexts, dimensions cannot be negative. Therefore, we need to ensure that both factors yield positive values for a given value of x. Generally, in such problems, x is assumed to be a value that makes both dimensions positive. Thus, the length and width of the rectangle can be represented as: Length = $(3x + 2)$ and Width = $(x - 5)$. These expressions indicate how the dimensions of the rectangle vary depending on the value of x. For example, if x is 6, the length would be $(3 \times 6 + 2) = 20$ units, and the width would be $(6 - 5) = 1$ unit. This solution demonstrates the practical application of factoring in geometric problems. By factoring the quadratic expression representing the area, Logan has successfully found the algebraic expressions for the length and width of the rectangle. This not only solves the problem but also highlights the interconnectedness of algebra and geometry.

Verification and Interpretation Ensuring a Valid Solution

After finding the dimensions of the rectangle, it is crucial to verify the solution to ensure its accuracy and validity. To verify, we can multiply the factors we obtained, $(x - 5)$ and $(3x + 2)$, and check if the result matches the original quadratic expression, $3x^2 - 13x - 10$. Multiplying the factors: $(x - 5)(3x + 2) = x(3x + 2) - 5(3x + 2) = 3x^2 + 2x - 15x - 10 = 3x^2 - 13x - 10$. The result matches the original quadratic expression, confirming that our factoring is correct. This verification step is essential in mathematics to avoid errors and build confidence in the solution. In addition to verifying the factoring, we must also interpret the solution in the context of the problem. Since the dimensions of a rectangle cannot be negative, we need to consider the implications of the factors $(x - 5)$ and $(3x + 2)$. For the width $(x - 5)$ to be positive, x must be greater than 5. Similarly, for the length $(3x + 2)$ to be positive, x must be greater than $-\frac{2}{3}$. Combining these conditions, x must be greater than 5 for both dimensions to be positive. This interpretation provides a complete understanding of the solution and its limitations. By verifying and interpreting the solution, Logan can be confident that he has not only found the correct factors but also understood the practical constraints of the problem. This holistic approach is vital in mathematical problem-solving.

Common Mistakes and How to Avoid Them

Factoring quadratic expressions can be challenging, and there are several common mistakes that students often make. Recognizing these pitfalls and understanding how to avoid them is crucial for mastering factoring. One common mistake is incorrectly identifying the factors of the product ac that add up to b. For example, in the expression $3x^2 - 13x - 10$, students might struggle to find the numbers -15 and 2. To avoid this, it is helpful to systematically list out the factor pairs of ac and check their sums. Another mistake is incorrectly factoring out the greatest common factor (GCF) from the grouped terms. Ensure that the GCF is factored out completely, leaving the correct expression inside the parentheses. For instance, in the step $(3x^2 - 15x) + (2x - 10)$, failing to factor out 3x from the first group or 2 from the second group would lead to incorrect factoring. A third common error is forgetting to factor out the common binomial factor in the final step. After factoring by grouping, both terms should have a common binomial factor. For example, after obtaining $3x(x - 5) + 2(x - 5)$, the common factor $(x - 5)$ must be factored out to get the final factored form. To avoid this, always double-check that the binomial factors are the same before factoring them out. Additionally, sign errors are frequent mistakes in factoring. Pay close attention to the signs of the terms when rewriting the middle term and factoring out the GCF. By being aware of these common mistakes and implementing strategies to avoid them, Logan can improve his factoring skills and achieve greater accuracy. Consistent practice and careful attention to detail are key to mastering factoring quadratic expressions.

Conclusion Mastering Factoring for Problem Solving

In conclusion, this exercise demonstrates the practical application of factoring quadratic expressions in solving geometric problems. Logan's task of finding the dimensions of a rectangle with an area of $3x^2 - 13x - 10$ required a thorough understanding of factoring techniques. By systematically factoring the quadratic expression, we identified the length and width of the rectangle as $(3x + 2)$ and $(x - 5)$, respectively. This process highlights the interconnectedness of algebra and geometry, showcasing how algebraic concepts can be used to solve real-world problems. The ability to factor quadratic expressions is a fundamental skill in algebra, with applications in various fields, including engineering, physics, and computer science. Mastering this skill requires a solid understanding of the underlying principles, attention to detail, and consistent practice. Furthermore, verifying the solution and interpreting it within the context of the problem are essential steps in ensuring accuracy and gaining a deeper understanding. By avoiding common mistakes and employing a systematic approach, students can confidently tackle factoring problems and apply this knowledge to a wide range of mathematical challenges. Logan's journey through this problem serves as a valuable lesson in the importance of perseverance, careful execution, and the practical relevance of algebraic skills. Ultimately, mastering factoring empowers students to approach complex problems with confidence and creativity.