Factoring P^2 - 20P + 51: A Simple Guide

Let's dive into factoring the quadratic expression P^2 - 20P + 51. Factoring is like reverse engineering multiplication – we're trying to find two expressions that, when multiplied together, give us the original expression. In this case, we want to find two binomials (expressions with two terms) that multiply to give us P^2 - 20P + 51. Understanding how to factor quadratic expressions is a fundamental skill in algebra, with applications ranging from solving equations to simplifying complex expressions. This particular quadratic expression is a trinomial, meaning it has three terms. The general form of a quadratic trinomial is ax^2 + bx + c, where a, b, and c are constants. In our case, a = 1, b = -20, and c = 51. Factoring this type of expression involves finding two numbers that add up to b and multiply to c. These numbers will then be used to construct the two binomial factors. This method is widely applicable to many quadratic expressions and serves as a cornerstone for more advanced algebraic techniques. Mastering this skill not only helps in solving equations but also enhances problem-solving abilities in various mathematical contexts.

Finding the Right Numbers

The key to factoring P^2 - 20P + 51 lies in finding two numbers that satisfy specific conditions. We need two numbers that, when added together, equal -20 (the coefficient of the P term), and when multiplied together, equal 51 (the constant term). Let's think about the factors of 51. Since 51 is positive and we need the numbers to add up to a negative number (-20), both numbers must be negative. The factors of 51 are 1 and 51, and 3 and 17. Considering the negative versions, we have -1 and -51, and -3 and -17. Now, let's check which pair adds up to -20. -1 + (-51) = -52, which is not -20. -3 + (-17) = -20. Bingo! We've found our numbers: -3 and -17. These two numbers are the magic ingredients that will allow us to rewrite and factor the expression. This process of identifying the correct factors is crucial, as it directly leads to the correct binomial factors. Without correctly identifying these numbers, the subsequent steps will be based on flawed information, leading to an incorrect factorization. This method can be systematically applied to any quadratic expression of the form x^2 + bx + c, making it a versatile tool in algebraic manipulations.

Constructing the Factored Form

Now that we've identified -3 and -17 as the numbers we need, we can construct the factored form of the expression P^2 - 20P + 51. The factored form will consist of two binomials, each containing P and one of the numbers we found. So, the factored form is (P - 3)(P - 17). This means that when you multiply (P - 3) by (P - 17), you should get back the original expression, P^2 - 20P + 51. It's always a good idea to check your work to make sure you factored correctly. This involves expanding the factored form using the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) and verifying that it simplifies to the original expression. Constructing the factored form is a straightforward process once the correct factors have been identified. The structure of the binomials directly reflects the factors, making it easy to translate the factors into the algebraic expression. This step is pivotal in solving quadratic equations and simplifying more complex algebraic expressions. A solid understanding of this process enables students to confidently tackle various factorization problems.

Verification: Expanding the Factored Form

To verify that (P - 3)(P - 17) is indeed the correct factored form of P^2 - 20P + 51, we need to expand the factored form and see if it simplifies back to the original expression. We can use the FOIL method: First: P * P = P^2 Outer: P * -17 = -17P Inner: -3 * P = -3P Last: -3 * -17 = 51 Now, let's add these terms together: P^2 - 17P - 3P + 51. Combining the like terms (-17P and -3P), we get P^2 - 20P + 51. This is exactly the original expression we started with! Therefore, our factored form (P - 3)(P - 17) is correct. The verification step is an essential part of the factoring process. It provides a check to ensure that the factorization is accurate and that no errors were made during the process. This practice reinforces the understanding of the relationship between factored and expanded forms of quadratic expressions. Moreover, it builds confidence in the problem-solving abilities of the individual, knowing that the answer has been thoroughly checked and confirmed. This step is crucial for ensuring accuracy and solidifying understanding of factoring techniques.

Why Factoring Matters

Factoring quadratic expressions like P^2 - 20P + 51 is not just an abstract mathematical exercise; it's a fundamental skill with wide-ranging applications in various fields. Factoring is essential for solving quadratic equations, which are used to model many real-world phenomena, such as the trajectory of a projectile, the area of a geometric shape, or the optimization of a business process. By factoring a quadratic equation, we can find its roots, which are the values of the variable that make the equation equal to zero. These roots often represent critical points or solutions to the problem being modeled. Moreover, factoring simplifies complex algebraic expressions, making them easier to work with and understand. It also plays a crucial role in calculus, where it is used to simplify expressions before differentiation or integration. In computer science, factoring is used in algorithms for cryptography and data compression. Therefore, mastering factoring techniques provides a solid foundation for advanced mathematical studies and practical problem-solving in various disciplines. It enhances analytical thinking and logical reasoning skills, which are valuable assets in any field. Understanding the importance of factoring motivates learners to engage with the material more deeply and appreciate its relevance beyond the classroom.

Conclusion

In summary, factoring the quadratic expression P^2 - 20P + 51 involves finding two numbers that add up to -20 and multiply to 51, which are -3 and -17. This allows us to write the expression in factored form as (P - 3)(P - 17). We verified this by expanding the factored form and confirming that it simplifies back to the original expression. Factoring is a crucial skill in algebra with numerous applications in mathematics, science, and engineering. By mastering factoring techniques, you'll be well-equipped to solve a wide range of problems and tackle more advanced mathematical concepts. Keep practicing and exploring different types of expressions to hone your factoring skills! For more information on factoring and other algebraic concepts, you can visit Khan Academy's Algebra Section.