Factor 3x² - 10x + 8 A Step-by-Step Guide To Quadratic Factorization

In the realm of algebra, factorization stands as a cornerstone skill, enabling us to break down complex expressions into simpler, more manageable components. Today, we embark on a journey to unravel the factorization of the quadratic expression 3x² - 10x + 8, a seemingly intricate puzzle that, with the right approach, can be elegantly solved. Our primary focus will be on completing the given factorization, which presents a partially factored form and challenges us to identify the missing piece. This process not only sharpens our factorization skills but also deepens our understanding of quadratic equations and their underlying structure.

Understanding the Factorization Process

Before we dive into the specifics of our problem, let's take a moment to appreciate the essence of factorization. At its core, factorization is the art of expressing a mathematical expression as a product of its factors. In the context of quadratic expressions, which take the general form of ax² + bx + c, factorization seeks to rewrite the expression as (px + q)(rx + s), where p, q, r, and s are constants. This transformation is incredibly valuable as it allows us to solve quadratic equations, simplify algebraic expressions, and gain insights into the behavior of quadratic functions.

The expression we are tasked with factorizing, 3x² - 10x + 8, is a classic example of a quadratic expression. The presence of the x² term signifies its quadratic nature, while the coefficients 3, -10, and 8 determine its specific characteristics. Our goal is to express this expression as a product of two binomials, which are expressions containing two terms. The partially factored form provided, (x - 2)(□x - 4), gives us a head start, revealing one of the binomial factors and leaving us to determine the missing coefficient in the other.

The Partially Factored Form: A Valuable Clue

The given partially factored form, (x - 2)(□x - 4), is a treasure trove of information. It tells us that one of the factors of our quadratic expression is (x - 2). This means that when x equals 2, the entire expression evaluates to zero. This is a crucial piece of information that can be used to verify our final factorization. Furthermore, the presence of the term (□x - 4) indicates that the other factor is also a binomial, with an unknown coefficient multiplying the x term.

Our mission is to fill in the blank, to determine the coefficient that, when multiplied by x, completes the factorization. To do this, we can employ several strategies, including the method of inspection, the quadratic formula, and the technique of splitting the middle term. Each approach offers a unique perspective on the factorization process, allowing us to choose the method that best suits our understanding and the specific problem at hand.

Method 1: The Art of Inspection

The method of inspection is a powerful tool that relies on our ability to recognize patterns and relationships within the expression. In this case, we can leverage our knowledge of how binomials multiply to deduce the missing coefficient. When we expand the product (x - 2)(□x - 4), we obtain a quadratic expression. The coefficient of the x² term in this expanded expression must match the coefficient of the x² term in our original expression, which is 3. Similarly, the constant term in the expanded expression must match the constant term in our original expression, which is 8.

Let's denote the missing coefficient as 'a'. When we expand (x - 2)(ax - 4), we get ax² - 4x - 2ax + 8. The coefficient of the x² term is 'a', and the constant term is 8. Since we know that the coefficient of the x² term in our original expression is 3, we can immediately conclude that a = 3. This gives us the factor (3x - 4). To verify our result, we can expand (x - 2)(3x - 4) and see if it matches our original expression. Expanding, we get 3x² - 4x - 6x + 8, which simplifies to 3x² - 10x + 8, confirming our factorization.

Method 2: Splitting the Middle Term

Another effective technique for factorization is splitting the middle term. This method involves rewriting the middle term (-10x in our case) as a sum of two terms such that the product of their coefficients equals the product of the coefficients of the x² term and the constant term (3 * 8 = 24). In other words, we need to find two numbers that add up to -10 and multiply to 24. These numbers are -6 and -4.

Now, we can rewrite our expression as 3x² - 6x - 4x + 8. We then group the terms in pairs: (3x² - 6x) + (-4x + 8). From the first group, we can factor out 3x, giving us 3x(x - 2). From the second group, we can factor out -4, giving us -4(x - 2). Notice that both groups now have a common factor of (x - 2). We can factor this out, resulting in (x - 2)(3x - 4), which is the same factorization we obtained using the method of inspection.

The Completed Factorization: A Triumph of Algebra

Through both the method of inspection and the technique of splitting the middle term, we have successfully completed the factorization of 3x² - 10x + 8. The missing coefficient in the partially factored form (x - 2)(□x - 4) is 3, giving us the complete factorization: (x - 2)(3x - 4). This factorization expresses the quadratic expression as a product of two binomials, revealing its underlying structure and providing a powerful tool for solving related equations and problems.

The Significance of Factorization

Factorization is not merely an algebraic trick; it is a fundamental skill with far-reaching applications. It allows us to solve quadratic equations, which are equations of the form ax² + bx + c = 0. By setting each factor in the factored form to zero, we can find the values of x that satisfy the equation, known as the roots or solutions. In our case, setting (x - 2) = 0 gives us x = 2, and setting (3x - 4) = 0 gives us x = 4/3. These are the two solutions to the quadratic equation 3x² - 10x + 8 = 0.

Furthermore, factorization simplifies algebraic expressions, making them easier to manipulate and understand. It also provides insights into the behavior of quadratic functions, which are functions of the form f(x) = ax² + bx + c. The roots of the quadratic equation, obtained through factorization, correspond to the x-intercepts of the quadratic function's graph, which is a parabola. The factored form also helps us identify the vertex of the parabola, which is the point where the function reaches its maximum or minimum value.

Mastering Factorization: A Journey of Practice and Discovery

Factorization, like any mathematical skill, requires practice and a willingness to explore different approaches. The more we engage with factorization problems, the more adept we become at recognizing patterns and applying the appropriate techniques. From simple quadratic expressions to more complex polynomials, factorization unlocks a world of algebraic possibilities. It empowers us to solve equations, simplify expressions, and gain a deeper understanding of mathematical relationships.

In conclusion, the factorization of 3x² - 10x + 8 into (x - 2)(3x - 4) is a testament to the power of algebraic techniques. By understanding the underlying principles of factorization and practicing various methods, we can confidently tackle a wide range of mathematical challenges. So, let us continue our journey of exploration and discovery, embracing the beauty and elegance of factorization.

Complete the factorization of the quadratic expression 3x² - 10x + 8, given that one factor is (x - 2). Find the missing factor in the form (□x - 4).

Factor 3x² - 10x + 8 A Step-by-Step Guide to Quadratic Factorization