Solving Consecutive Integer Problems Factoring The Equation X(x+1)=72
In the fascinating world of mathematics, we often encounter puzzles that require us to think critically and apply our knowledge of equations and algebraic principles. One such intriguing problem involves consecutive integers and their products. Let's delve into the depths of this problem, unraveling its complexities and discovering the elegant solutions it holds.
Understanding the Problem: Consecutive Integers and Their Products
At the heart of this problem lies the concept of consecutive integers. These are integers that follow each other in sequence, each differing from the previous one by exactly 1. Examples of consecutive integers include 5 and 6, -3 and -2, or even 100 and 101. The problem states that the product of two such consecutive integers is 72. Our mission is to decipher which equation, among the given options, can be factored and solved to pinpoint the smaller of these two elusive integers.
The equation x(x+1) = 72 serves as the compass guiding us through this mathematical maze. Here, 'x' represents the smaller integer, and 'x+1' naturally represents the next consecutive integer. The equation eloquently translates the problem's core statement: the product of these two consecutive integers is indeed 72. Now, the challenge lies in maneuvering this equation into a form that allows us to factorize it and extract the value of 'x', the smaller integer we seek.
Transforming the Equation: Setting the Stage for Factorization
The equation x(x+1) = 72, in its current guise, isn't immediately amenable to factorization. To unlock its hidden potential, we need to embark on a journey of algebraic transformation. Our first step involves expanding the left side of the equation. Distributing 'x' across the parentheses, we arrive at x² + x = 72. This quadratic expression is starting to take shape, but we're not quite there yet.
To fully prepare the equation for factorization, we need to bring all the terms to one side, leaving zero on the other side. This is achieved by subtracting 72 from both sides of the equation, resulting in the pivotal form: x² + x - 72 = 0. This equation, now in its standard quadratic form, stands poised for factorization. The stage is set, and the spotlight is on finding the factors that will unveil the value of 'x'.
Factorization Unveiled: Cracking the Code
Factorization is the art of dissecting a mathematical expression into its constituent factors – expressions that, when multiplied together, yield the original expression. In the case of our quadratic equation, x² + x - 72 = 0, we seek two binomial expressions that, when multiplied, produce this very equation. This involves identifying two numbers that, when multiplied, give us -72 (the constant term) and, when added, give us 1 (the coefficient of the 'x' term).
Through careful consideration, we discover that the numbers 9 and -8 fit this description perfectly. Their product is indeed -72, and their sum is 1. Armed with this knowledge, we can rewrite the quadratic equation in its factored form: (x + 9)(x - 8) = 0. This factorization is the key to unlocking the values of 'x' that satisfy the equation.
Solving for x: Unearthing the Solutions
The factored equation, (x + 9)(x - 8) = 0, presents us with a powerful insight. For the product of two expressions to be zero, at least one of them must be zero. This leads us to two distinct possibilities:
- x + 9 = 0
- x - 8 = 0
Solving the first equation, x + 9 = 0, we subtract 9 from both sides, revealing that x = -9. This is one potential value for the smaller integer.
Solving the second equation, x - 8 = 0, we add 8 to both sides, unearthing the solution x = 8. This is another possible value for the smaller integer.
The Final Verdict: Identifying the Smaller Integer
We have arrived at two potential solutions for the smaller integer: x = -9 and x = 8. Both values satisfy the equation x² + x - 72 = 0, but we must carefully consider the context of the problem to determine the correct answer. The problem asks for the smaller integer. Comparing the two solutions, -9 is indeed smaller than 8.
Therefore, the smaller integer is -9. This is the solution that aligns perfectly with the problem's conditions, completing our mathematical journey.
Exploring Alternative Equations and Factorization Techniques
While we have successfully navigated the problem using the equation x² + x - 72 = 0, it's worth noting that other equivalent equations could also lead us to the correct solution. For instance, we could have added 72 to both sides of the original equation, resulting in x² + x = 72. This form, while not immediately factorable, could be manipulated further to arrive at the standard quadratic form.
Furthermore, alternative factorization techniques could be employed. The quadratic formula, for instance, provides a general method for solving quadratic equations of the form ax² + bx + c = 0. While not strictly factorization, it offers a reliable route to finding the roots of the equation, which in our case, correspond to the potential values of the smaller integer.
The Significance of Consecutive Integer Problems
Problems involving consecutive integers are not mere mathematical exercises; they often serve as stepping stones to more complex algebraic concepts. They help us develop our skills in equation manipulation, factorization, and problem-solving strategies. Moreover, they provide a glimpse into the patterns and relationships that govern the world of numbers.
In various fields, from computer science to finance, the ability to work with sequences and patterns is crucial. Consecutive integer problems, therefore, contribute to building a foundation for these advanced applications.
Conclusion: The Power of Factorization
In this exploration of consecutive integers and their products, we have witnessed the power of factorization as a problem-solving tool. By transforming the equation x(x+1) = 72 into its factored form, (x + 9)(x - 8) = 0, we were able to unearth the solutions for the smaller integer. The journey underscored the importance of algebraic manipulation, careful consideration of the problem's context, and the versatility of factorization techniques.
As we continue our mathematical pursuits, let's remember the lessons learned from this endeavor. Consecutive integer problems, with their blend of algebraic elegance and practical applications, offer a valuable opportunity to hone our problem-solving skills and deepen our appreciation for the beauty of mathematics.
Which equation can be factored and solved for the smaller integer, given that the product of two consecutive integers is 72 and represented by the equation x(x+1)=72, where x is the smaller integer?
Solving Consecutive Integer Problems Factoring the Equation x(x+1)=72
