Factorising X² - 15x + 56 A Step By Step Guide
Factorising quadratic expressions is a fundamental skill in algebra, and it's essential for solving equations, simplifying expressions, and understanding various mathematical concepts. In this article, we will delve into the process of factorising a specific quadratic expression: x² - 15x + 56. We will break down the steps involved, explain the underlying principles, and provide insights into why this skill is crucial in mathematics. By the end of this guide, you'll have a solid understanding of how to factorise quadratic expressions and confidently tackle similar problems.
Understanding Quadratic Expressions
Before we dive into the factorisation process, it's crucial to understand what quadratic expressions are and their general form. A quadratic expression is a polynomial expression of degree two, meaning the highest power of the variable (usually 'x') is two. The general form of a quadratic expression is ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our case, the expression x² - 15x + 56 fits this form, with a = 1, b = -15, and c = 56.
Why are quadratic expressions so important? They appear in various mathematical contexts, including solving quadratic equations, graphing parabolas, and modelling real-world phenomena such as projectile motion and optimization problems. Mastering the art of factorising quadratic expressions opens the door to a deeper understanding of these concepts and their applications. Understanding the structure of a quadratic expression is the first step towards successfully factorising it. Recognizing the coefficients 'a', 'b', and 'c' helps in applying the appropriate factorisation techniques. In our example, the coefficient of x² is 1, the coefficient of x is -15, and the constant term is 56. These values will guide us as we navigate the factorisation process. Remember that factorising is essentially the reverse process of expanding brackets. When we expand (x + p)(x + q), we get x² + (p + q)x + pq. Therefore, when factorising, we are looking for two numbers, p and q, that add up to the coefficient of x (which is 'b') and multiply to the constant term (which is 'c'). This connection between expansion and factorisation is a cornerstone of algebraic manipulation. Furthermore, quadratic expressions are closely linked to quadratic equations. A quadratic equation is formed when a quadratic expression is set equal to zero (ax² + bx + c = 0). The solutions to this equation, also known as the roots, can often be found by factorising the quadratic expression and setting each factor equal to zero. This technique is one of the most powerful tools for solving quadratic equations, making factorisation a core skill in algebra.
The Factorisation Process: A Step-by-Step Approach
Now, let's break down the process of factorising x² - 15x + 56 into manageable steps. The key to factorising quadratic expressions lies in finding two numbers that satisfy specific conditions related to the coefficients of the expression. In this case, we need to find two numbers that add up to -15 (the coefficient of x) and multiply to 56 (the constant term). This step is crucial as it sets the foundation for the entire factorisation process. Without identifying these numbers correctly, the subsequent steps will not lead to the correct factorisation. There are several strategies for finding these numbers, including trial and error, listing factor pairs, and using the quadratic formula (although the latter is more commonly used for solving equations rather than factorising). For this expression, we'll focus on listing factor pairs and using logical deduction. Once we've identified the correct numbers, we can rewrite the middle term (-15x) as the sum of two terms using these numbers. This step transforms the three-term expression into a four-term expression, which allows us to apply the grouping method for factorisation. The grouping method involves pairing the terms and factoring out the greatest common factor (GCF) from each pair. This process will reveal a common binomial factor, which can then be factored out, resulting in the final factorised form of the quadratic expression. By following these steps systematically, we can effectively factorise a wide range of quadratic expressions.
Step 1: Finding the Right Numbers
The first crucial step is to identify two numbers that meet our criteria: they must add up to -15 and multiply to 56. This is where our understanding of number properties and a bit of strategic thinking come into play. We need to consider the factors of 56 and see which pair, when combined with appropriate signs, will give us a sum of -15. Let's list the factor pairs of 56: (1, 56), (2, 28), (4, 14), and (7, 8). Since we need the numbers to multiply to a positive value (56) but add up to a negative value (-15), both numbers must be negative. This narrows down our possibilities significantly. Now, let's consider the negative counterparts of these pairs: (-1, -56), (-2, -28), (-4, -14), and (-7, -8). Adding these pairs, we get -57, -30, -18, and -15, respectively. Aha! The pair -7 and -8 satisfy both conditions: -7 + (-8) = -15 and -7 * -8 = 56. This careful analysis of factor pairs and their properties is a key skill in factorisation. It allows us to efficiently narrow down the possibilities and identify the numbers that will lead to the correct factorisation. Remember, practice makes perfect, and with experience, you'll become more adept at recognizing these number patterns.
Step 2: Rewriting the Middle Term
Now that we've found the numbers -7 and -8, we can rewrite the middle term of our expression, -15x, as the sum of -7x and -8x. This seemingly simple step is a crucial bridge between the original quadratic expression and its factorised form. By rewriting the middle term, we transform the three-term expression (x² - 15x + 56) into a four-term expression (x² - 7x - 8x + 56). This transformation is not just cosmetic; it sets the stage for applying the grouping method, a powerful technique for factorising expressions with four terms. The key idea behind rewriting the middle term is to create two pairs of terms that share a common factor. This common factor will then be factored out in the next step, leading to the identification of a common binomial factor. In our case, the four-term expression x² - 7x - 8x + 56 allows us to group the first two terms (x² - 7x) and the last two terms (-8x + 56). Each of these groups will reveal a common factor, which will ultimately lead us to the factorised form of the original expression. This strategic rewriting of the middle term is a fundamental technique in factorisation and is widely applicable to various quadratic expressions.
Step 3: Factorising by Grouping
With our expression rewritten as x² - 7x - 8x + 56, we can now employ the factorising by grouping method. This technique involves grouping the first two terms and the last two terms and then factoring out the greatest common factor (GCF) from each group. Let's start with the first group: x² - 7x. The GCF of these two terms is x. Factoring out x, we get x(x - 7). Now, let's move on to the second group: -8x + 56. The GCF of these two terms is -8 (we factor out -8 to ensure the binomial factor matches the one from the first group). Factoring out -8, we get -8(x - 7). Notice that we now have two terms: x(x - 7) and -8(x - 7). Both terms share a common binomial factor: (x - 7). This is the key outcome of the grouping method. We can now factor out this common binomial factor, treating it as a single unit. Factoring out (x - 7) from the entire expression, we get (x - 7)(x - 8). This is the factorised form of our quadratic expression. The beauty of the grouping method lies in its systematic approach. By carefully grouping terms and factoring out GCFs, we reveal the underlying structure of the expression and arrive at its factorised form. This technique is particularly useful for expressions with four terms and is a cornerstone of algebraic manipulation.
The Solution: (x - 7)(x - 8)
Therefore, the factorised form of x² - 15x + 56 is (x - 7)(x - 8). This corresponds to option D in the given choices. We have successfully navigated the factorisation process, starting with understanding the structure of quadratic expressions, identifying the key numbers, rewriting the middle term, and applying the grouping method. This step-by-step approach provides a clear and logical path to factorising quadratic expressions. It's important to note that factorisation is not just about arriving at the correct answer; it's about understanding the underlying principles and developing problem-solving skills. By mastering the techniques involved in factorising, you gain a deeper appreciation for the interconnectedness of algebraic concepts and enhance your ability to tackle more complex mathematical challenges. Remember to always double-check your answer by expanding the factors to ensure you arrive back at the original expression. In this case, expanding (x - 7)(x - 8) gives us x² - 8x - 7x + 56, which simplifies to x² - 15x + 56, confirming our factorisation is correct.
Why is Factorising Important?
Factorising quadratic expressions is not just an isolated algebraic skill; it's a fundamental tool that underpins various mathematical concepts and applications. Factorisation is crucial for solving quadratic equations, which are equations of the form ax² + bx + c = 0. By factorising the quadratic expression, we can rewrite the equation as a product of two linear factors, each of which can be set equal to zero to find the solutions (also known as roots) of the equation. This method is often more efficient than using the quadratic formula, especially when the quadratic expression can be easily factorised. Beyond solving equations, factorisation plays a vital role in simplifying algebraic expressions. Complex expressions can often be simplified by factorising common factors and cancelling them out. This simplification process is essential for performing operations such as adding, subtracting, multiplying, and dividing algebraic fractions. In calculus, factorisation is used to find the roots of polynomial functions, which are crucial for determining the critical points and intervals of increase and decrease of the function. These concepts are fundamental to understanding the behaviour of functions and solving optimization problems. Furthermore, factorisation has applications in various real-world scenarios, such as modelling projectile motion, designing structures, and optimizing business processes. Quadratic equations and their solutions are used to model the trajectory of objects, calculate the load-bearing capacity of beams, and determine the optimal pricing strategies for products. By mastering the skill of factorising, you unlock a powerful toolset for tackling a wide range of mathematical problems and real-world applications.
Practice Makes Perfect: Additional Tips and Exercises
To truly master factorising quadratic expressions, consistent practice is key. Start with simple expressions and gradually work your way up to more complex ones. Practice identifying the pairs of numbers that add up to the coefficient of x and multiply to the constant term. This skill is the cornerstone of successful factorisation. Use various techniques, such as listing factor pairs and applying logical deduction, to find these numbers efficiently. When you encounter expressions with a leading coefficient (a ≠ 1), the factorisation process becomes slightly more complex, but the underlying principles remain the same. You'll need to consider the factors of both the leading coefficient and the constant term and experiment with different combinations to find the correct factors. Don't be afraid to make mistakes; they are valuable learning opportunities. When you get stuck, review the steps involved in the factorisation process and try to identify where you might have gone wrong. There are also numerous online resources and textbooks that provide additional examples and practice problems. Work through these resources and seek help from teachers or tutors when needed. Remember, the more you practice, the more confident and proficient you'll become in factorising quadratic expressions. To further enhance your understanding, try factorising the following expressions:
- x² + 5x + 6
- x² - 8x + 15
- x² + 2x - 8
- 2x² + 7x + 3
- 3x² - 5x - 2
By working through these exercises, you'll solidify your understanding of the factorisation process and develop the skills needed to tackle a wide range of quadratic expressions. Remember to always check your answers by expanding the factors to ensure you arrive back at the original expression.
Conclusion: Mastering the Art of Factorisation
In conclusion, factorising quadratic expressions is a fundamental skill in algebra with wide-ranging applications. In this article, we've walked through a step-by-step guide to factorising the expression x² - 15x + 56, highlighting the key concepts and techniques involved. We've emphasised the importance of understanding the structure of quadratic expressions, identifying the correct numbers, rewriting the middle term, and applying the grouping method. By mastering these steps, you can confidently factorise a wide variety of quadratic expressions. We've also discussed the importance of factorisation in solving quadratic equations, simplifying algebraic expressions, and understanding various mathematical concepts. Factorisation is not just a mathematical skill; it's a problem-solving tool that enhances your analytical thinking and logical reasoning abilities. Consistent practice is crucial for mastering factorisation. Work through numerous examples, seek help when needed, and don't be afraid to make mistakes. Each mistake is an opportunity to learn and grow. With dedication and perseverance, you can master the art of factorisation and unlock a deeper understanding of algebra and its applications. Remember to always check your answers and strive for a conceptual understanding of the underlying principles. Happy factorising!
