Complete Factorization Of 5x² - 11x - 12 A Step-by-Step Guide

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Finding the complete factorization of a quadratic expression like 5x² - 11x - 12 is a fundamental skill in algebra. It allows us to simplify expressions, solve equations, and understand the behavior of quadratic functions. This article delves deep into the process of factoring this specific quadratic, providing a step-by-step guide and explaining the underlying principles. We'll explore the techniques involved, from identifying the coefficients to strategically splitting the middle term, ensuring you grasp the complete factorization process. The ability to factor quadratics is not just crucial for academic success in mathematics, but it also has applications in various fields, including physics, engineering, and computer science. Mastering this skill will undoubtedly enhance your problem-solving capabilities and provide a solid foundation for more advanced mathematical concepts. This detailed exploration aims to equip you with the knowledge and confidence to tackle similar factorization problems with ease. So, let's embark on this journey of algebraic discovery and unlock the secrets hidden within the expression 5x² - 11x - 12. We will break down each step, making it clear and understandable, so you can apply these techniques to other quadratic expressions you encounter. Remember, practice is key to mastering any mathematical skill, so as you follow along, try working through similar examples to solidify your understanding. By the end of this article, you will not only know the complete factorization of 5x² - 11x - 12 but also have a robust understanding of the factoring process itself.

Understanding the Quadratic Expression

Before we jump into the factorization process, let's break down the quadratic expression 5x² - 11x - 12. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, x) is 2. The general form of a quadratic expression is ax² + bx + c, where a, b, and c are constants. In our specific expression, we can identify the coefficients as follows:

  • a = 5 (the coefficient of x²)
  • b = -11 (the coefficient of x)
  • c = -12 (the constant term)

These coefficients play a crucial role in determining the factors of the quadratic expression. The goal of factorization is to rewrite the quadratic expression as a product of two linear expressions, which have the form (px + q), where p and q are constants. In essence, we are trying to find two binomials that, when multiplied together, will give us the original quadratic expression. This process often involves trial and error, but there are systematic methods that can help us find the correct factors more efficiently. One of the most common methods is the "splitting the middle term" technique, which we will explore in detail later. Understanding the relationship between the coefficients and the factors is essential for mastering factorization. It allows us to make educated guesses and narrow down the possibilities, making the process less daunting. So, before we move on to the actual factorization steps, make sure you have a clear understanding of the roles played by the coefficients a, b, and c. This foundational knowledge will greatly aid in your understanding and application of the factorization techniques we will discuss.

The Splitting the Middle Term Method

The "splitting the middle term" method is a powerful technique for factoring quadratic expressions. It involves rewriting the middle term (bx) as the sum of two terms, such that the resulting expression can be factored by grouping. The key to this method lies in finding two numbers that satisfy specific conditions related to the coefficients a, b, and c. Let's outline the steps involved in this method for our expression, 5x² - 11x - 12:

  1. Calculate ac: Multiply the coefficient of x² (a) by the constant term (c). In our case, ac = 5 * (-12) = -60.
  2. Find two numbers: Find two numbers that multiply to ac (-60) and add up to the coefficient of x (b, which is -11). This is the crucial step, and it might require some trial and error. We need to think of factor pairs of -60 and see which pair adds up to -11. The pairs could be (1, -60), (-1, 60), (2, -30), (-2, 30), and so on. After considering the possibilities, we find that the numbers 4 and -15 satisfy the conditions because 4 * (-15) = -60 and 4 + (-15) = -11.
  3. Split the middle term: Rewrite the middle term (-11x) as the sum of the two terms using the numbers we found (4 and -15). So, -11x becomes 4x - 15x. Our expression now becomes 5x² + 4x - 15x - 12.
  4. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair. From the first two terms, 5x² + 4x, the GCF is x, so we factor it out: x(5x + 4). From the last two terms, -15x - 12, the GCF is -3, so we factor it out: -3(5x + 4). Our expression now looks like x(5x + 4) - 3(5x + 4).
  5. Factor out the common binomial: Notice that both terms now have a common binomial factor, which is (5x + 4). Factor this out: (5x + 4)(x - 3).This completes the factorization process. We have successfully rewritten the quadratic expression as a product of two linear expressions.

Step-by-Step Factorization of 5x² - 11x - 12

Now, let's walk through the complete factorization of 5x² - 11x - 12 using the splitting the middle term method, consolidating the steps we discussed earlier:

  1. Calculate ac: As we determined before, a = 5 and c = -12, so ac = 5 * (-12) = -60.
  2. Find two numbers: We need two numbers that multiply to -60 and add up to -11. After considering the factor pairs of -60, we identify the numbers 4 and -15. Indeed, 4 * (-15) = -60 and 4 + (-15) = -11.
  3. Split the middle term: Rewrite -11x as 4x - 15x. The expression becomes: 5x² + 4x - 15x - 12.
  4. Factor by grouping: Group the terms in pairs: (5x² + 4x) + (-15x - 12). Factor out the GCF from each pair:
    • From 5x² + 4x, the GCF is x, so we get x(5x + 4).
    • From -15x - 12, the GCF is -3, so we get -3(5x + 4).The expression now looks like: x(5x + 4) - 3(5x + 4).
  5. Factor out the common binomial: Notice the common binomial factor (5x + 4). Factor it out: (5x + 4)(x - 3).Therefore, the complete factorization of 5x² - 11x - 12 is (5x + 4)(x - 3). This step-by-step breakdown demonstrates the methodical approach of the splitting the middle term method. Each step builds upon the previous one, leading us to the final factored form. It's important to practice these steps with different quadratic expressions to become proficient in factorization. The ability to factor quadratics quickly and accurately is a valuable asset in various mathematical contexts. So, take the time to understand each step and apply it to a variety of problems to solidify your skills.

The Complete Factorization

After diligently applying the splitting the middle term method, we have arrived at the complete factorization of the quadratic expression 5x² - 11x - 12. The factored form is (5x + 4)(x - 3). This means that if we were to multiply these two binomials together, we would obtain the original quadratic expression. Let's verify this by expanding the product:

(5x + 4)(x - 3) = 5x(x - 3) + 4(x - 3)

Now, distribute the terms:

= 5x² - 15x + 4x - 12

Combine like terms:

= 5x² - 11x - 12

As we can see, expanding the factored form (5x + 4)(x - 3) indeed gives us the original expression 5x² - 11x - 12, confirming that our factorization is correct. The complete factorization provides valuable insights into the roots (or zeros) of the corresponding quadratic equation 5x² - 11x - 12 = 0. The roots are the values of x that make the equation true. In factored form, it becomes clear that the roots are the values that make each factor equal to zero. Setting each factor to zero, we get:

  • 5x + 4 = 0 => x = -4/5
  • x - 3 = 0 => x = 3Thus, the roots of the quadratic equation are x = -4/5 and x = 3. This demonstrates the power of factorization in solving quadratic equations. By expressing the quadratic in factored form, we can easily identify its roots. Understanding the complete factorization not only simplifies algebraic manipulations but also provides a deeper understanding of the underlying mathematical relationships. It's a fundamental skill that opens doors to more advanced concepts in algebra and beyond. So, mastering this process is crucial for anyone pursuing mathematics or related fields.

Therefore, the complete factorization of 5x² - 11x - 12 is (5x + 4)(x - 3).