Complete Factorization Of 4x² - 26x + 30 A Step-by-Step Guide
In this article, we will delve into the process of completely factoring the quadratic expression 4x² - 26x + 30. Factoring quadratic expressions is a fundamental skill in algebra, and mastering it opens doors to solving quadratic equations, simplifying algebraic fractions, and tackling more advanced mathematical concepts. We'll break down the steps involved, ensuring a clear understanding of each stage. This exploration aims to provide not just the solution but a comprehensive understanding of the underlying principles of factorization.
Factoring Quadratic Expressions: A Step-by-Step Guide
Factoring quadratic expressions is a crucial skill in algebra, and it involves breaking down a quadratic expression into a product of simpler expressions, usually binomials. This process is essentially the reverse of expanding brackets, and it relies on identifying common factors and recognizing patterns. Factoring allows us to solve quadratic equations, simplify algebraic expressions, and understand the behavior of quadratic functions. The expression we are tackling today, 4x² - 26x + 30, is a standard quadratic trinomial, and we will employ a methodical approach to factorize it completely. The goal is to rewrite this expression as a product of two binomials, and to do so effectively, we'll need to consider the coefficients, constants, and the signs within the expression. This involves not only mathematical manipulation but also a strategic application of algebraic principles to ensure we arrive at the correct factored form. Furthermore, the ability to factor quadratic expressions proficiently is not just a standalone skill; it is a building block for more complex mathematical operations and problem-solving scenarios. The techniques learned here can be applied in various contexts, from calculus to engineering, making it a cornerstone of mathematical education.
Step 1: Identifying the Greatest Common Factor (GCF)
The initial step in completely factoring any algebraic expression, including our quadratic 4x² - 26x + 30, is to identify the Greatest Common Factor (GCF) of all the terms. The GCF is the largest factor that divides each term in the expression without leaving a remainder. In our case, we have three terms: 4x², -26x, and +30. Examining the coefficients (4, -26, and 30), we can see that the largest number that divides all three is 2. There is no common variable factor since the constant term, 30, does not contain the variable 'x'. Therefore, the GCF of the entire expression is 2. Factoring out the GCF simplifies the expression and makes subsequent factoring steps easier. By dividing each term by 2, we transform the original expression into a simpler form that is more manageable to work with. This step is crucial because it ensures that the final factored form is in its simplest terms, and it also helps to avoid errors in the later stages of factorization. The importance of identifying the GCF cannot be overstated, as it is often the key to unlocking the complete factorization of a polynomial expression. Neglecting this step can lead to incomplete factorization or more complex factoring procedures.
Step 2: Factoring out the GCF
Having identified the GCF as 2, we now proceed to factor it out from the expression 4x² - 26x + 30. This involves dividing each term of the expression by the GCF and rewriting the expression as the product of the GCF and the resulting quotient. When we divide 4x² by 2, we obtain 2x². Dividing -26x by 2 gives us -13x, and dividing +30 by 2 yields +15. Thus, the expression 4x² - 26x + 30 can be rewritten as 2(2x² - 13x + 15). This step is a critical transformation because it reduces the complexity of the quadratic expression inside the parentheses, making it easier to factor further. By factoring out the GCF, we have essentially extracted a common multiplier from the entire expression, which simplifies the task of finding the binomial factors. The new expression, 2x² - 13x + 15, is a simpler quadratic trinomial that we can now focus on factoring. This process highlights the efficiency of factoring out the GCF as a first step in simplifying algebraic expressions and setting the stage for subsequent factorization techniques. The correct application of this step is fundamental to achieving a complete and accurate factorization.
Step 3: Factoring the Remaining Quadratic Expression
Now, we focus on factoring the quadratic expression inside the parentheses, which is 2x² - 13x + 15. This is a trinomial of the form ax² + bx + c, where a = 2, b = -13, and c = 15. To factor this type of quadratic, we need to find two numbers that multiply to ac (2 * 15 = 30) and add up to b (-13). This is a classic technique in factoring quadratics, often referred to as the 'ac method'. We systematically consider pairs of factors of 30 and check if their sum or difference can give us -13. After exploring the factor pairs, we find that the numbers -10 and -3 satisfy these conditions: (-10) * (-3) = 30 and (-10) + (-3) = -13. These numbers are the key to rewriting the middle term (-13x) of the quadratic expression. We rewrite -13x as -10x - 3x, which allows us to factor by grouping. This technique involves splitting the middle term and then factoring pairs of terms within the expression. The ability to identify the correct numbers that satisfy the multiplication and addition conditions is crucial for successful factoring. It requires a good understanding of number properties and the ability to apply them effectively in an algebraic context.
Step 4: Factoring by Grouping
Having identified the numbers -10 and -3, we rewrite the quadratic expression 2x² - 13x + 15 as 2x² - 10x - 3x + 15. This step prepares the expression for factoring by grouping, a technique that involves pairing terms and extracting common factors from each pair. We group the first two terms and the last two terms: (2x² - 10x) + (-3x + 15). Now, we factor out the GCF from each group. From the first group, 2x² - 10x, the GCF is 2x. Factoring out 2x gives us 2x(x - 5). From the second group, -3x + 15, the GCF is -3. Factoring out -3 gives us -3(x - 5). Notice that both groups now have a common binomial factor of (x - 5). This is a critical point in the factoring process, as it indicates that we are on the right track. The next step is to factor out this common binomial factor. By factoring out (x - 5) from the entire expression, we are essentially reversing the distributive property. This leads us to the factored form of the quadratic expression. The skill of factoring by grouping is a powerful tool in algebra, applicable not only to quadratics but also to polynomials of higher degrees. It relies on careful observation and the ability to recognize common factors within different parts of an expression.
Step 5: Completing the Factorization
After factoring by grouping, we have the expression 2x(x - 5) - 3(x - 5). As observed earlier, both terms share a common binomial factor of (x - 5). We now factor out this common binomial factor, which results in (x - 5)(2x - 3). This represents the factored form of the quadratic expression 2x² - 13x + 15. However, we must remember the GCF that we factored out in the initial step. We need to include this GCF in the final factorization to obtain the complete factored form of the original expression, 4x² - 26x + 30. Therefore, we multiply the factored quadratic expression by the GCF, which was 2. This gives us the complete factored form: 2(x - 5)(2x - 3). This is the final result of our factoring process. It represents the original quadratic expression as a product of a constant and two binomials. The process of completing the factorization involves not only correctly factoring the quadratic but also ensuring that all common factors are accounted for. This step demonstrates the importance of revisiting earlier steps in the factoring process to ensure that the final result is complete and accurate. The ability to factor expressions completely is a crucial skill in algebra and has wide-ranging applications in various mathematical contexts.
Solution and Verification
Therefore, the complete factorization of 4x² - 26x + 30 is 2(x - 5)(2x - 3). This corresponds to option B. To verify our solution, we can expand the factored form and check if it matches the original expression. Expanding 2(x - 5)(2x - 3), we first multiply the binomials (x - 5) and (2x - 3). This gives us 2x² - 3x - 10x + 15, which simplifies to 2x² - 13x + 15. Next, we multiply this result by the GCF, 2, which gives us 4x² - 26x + 30, the original expression. This confirms that our factorization is correct. The verification step is an essential part of the factoring process. It provides a check on our work and ensures that we have not made any errors in the factoring process. By expanding the factored form, we reverse the factoring process and should arrive back at the original expression. If the expanded form does not match the original expression, it indicates an error in the factoring process, and we need to review our steps. The ability to verify solutions is a crucial skill in mathematics and is essential for building confidence in one's work.
Conclusion
In conclusion, we have successfully factored the quadratic expression 4x² - 26x + 30 completely. The step-by-step approach, involving identifying the GCF, factoring out the GCF, factoring the remaining quadratic expression, factoring by grouping, and completing the factorization, has provided a clear and methodical way to arrive at the solution. The factored form, 2(x - 5)(2x - 3), not only represents the original expression in a different form but also provides valuable insights into the roots and behavior of the corresponding quadratic equation. Furthermore, the process of verifying the solution by expanding the factored form has reinforced the accuracy of our work. Mastering the techniques of factoring quadratic expressions is essential for success in algebra and beyond. These skills are fundamental for solving equations, simplifying expressions, and understanding mathematical relationships. The ability to factor efficiently and accurately is a valuable asset in any mathematical endeavor. The comprehensive understanding of factorization gained through this exercise will serve as a strong foundation for tackling more complex algebraic problems in the future.
