Solving Y^2 + 7y = 18 By Factoring A Step By Step Guide
In the realm of mathematics, quadratic equations hold a significant place, appearing in various fields ranging from physics to engineering. Mastering the techniques to solve these equations is crucial for any aspiring mathematician or scientist. One of the most fundamental methods for solving quadratic equations is factoring. In this article, we will delve into the process of solving the quadratic equation y^2 + 7y = 18 by factoring, providing a step-by-step guide and exploring the underlying principles. This equation serves as a classic example to illustrate how factoring can lead to a straightforward solution. We will dissect the equation, rearrange it into a standard form, and then factorize it, ultimately identifying the values of y that satisfy the equation. Understanding the nuances of factoring quadratic equations not only enhances problem-solving skills but also lays a solid foundation for more advanced mathematical concepts. Therefore, whether you are a student grappling with algebra or simply a math enthusiast seeking to refresh your knowledge, this guide will offer valuable insights into the world of quadratic equations and the art of factoring. We will explore how to rearrange the equation into its standard form, identify the key components that allow us to factor it effectively, and then apply the factoring technique to find the roots. This approach will not only solve the given equation but also provide a template for tackling similar problems in the future. The ability to solve quadratic equations by factoring is a cornerstone of algebraic proficiency, and this article aims to make that skill accessible to all.
Understanding Quadratic Equations
To effectively tackle the equation y^2 + 7y = 18, it's crucial to first grasp the concept of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. These constants play a critical role in determining the nature and solutions of the equation. The coefficient a dictates the parabola's direction (whether it opens upwards or downwards) and its steepness, while b influences the parabola's position along the x-axis, and c represents the y-intercept. Understanding these coefficients is essential for visualizing and manipulating quadratic equations. The solutions to a quadratic equation, also known as roots or zeros, are the values of the variable that make the equation true. These roots represent the points where the parabola intersects the x-axis. A quadratic equation can have two distinct real roots, one real root (when the parabola touches the x-axis at only one point), or no real roots (when the parabola does not intersect the x-axis). There are several methods to find these roots, including factoring, using the quadratic formula, and completing the square. Each method has its own advantages and is suitable for different types of quadratic equations. For instance, factoring is most effective when the equation can be easily decomposed into two binomials, while the quadratic formula provides a universal solution regardless of the equation's complexity. Completing the square is a powerful technique that not only solves equations but also transforms them into a vertex form, revealing the vertex of the parabola. In the context of y^2 + 7y = 18, we will focus on the factoring method, which is particularly efficient when the equation can be expressed as a product of two binomials. By understanding the structure and properties of quadratic equations, we can approach problem-solving with greater confidence and accuracy.
Step 1: Rearranging the Equation
The first crucial step in solving the quadratic equation y^2 + 7y = 18 by factoring is to rearrange it into the standard quadratic form, which is ay^2 + by + c = 0. This form is essential because it sets the stage for the factoring process by organizing all the terms on one side of the equation and setting the other side to zero. To achieve this, we need to subtract 18 from both sides of the equation. This operation ensures that the equation remains balanced, a fundamental principle in algebraic manipulations. Subtracting 18 from both sides of y^2 + 7y = 18 yields: y^2 + 7y - 18 = 0. Now, the equation is in the standard quadratic form, where a = 1, b = 7, and c = -18. These coefficients are critical for the next steps in the factoring process. The standard form not only provides a clear structure but also allows us to easily identify the coefficients, which are necessary for applying various solving techniques, including factoring and the quadratic formula. Moreover, rearranging the equation into the standard form is a common practice in solving various types of equations, not just quadratic ones. It helps in simplifying the problem and making it more manageable. By having all terms on one side and zero on the other, we create a clear picture of the equation's structure, which is crucial for determining the most appropriate solution method. In the case of factoring, the standard form allows us to systematically look for two binomials that, when multiplied, will give us the quadratic expression. This process involves finding two numbers that add up to the coefficient of the linear term (b) and multiply to the constant term (c). Therefore, mastering the skill of rearranging equations into standard form is a fundamental step in algebra and a prerequisite for solving a wide range of mathematical problems.
Step 2: Factoring the Quadratic Expression
With the equation now in standard form, y^2 + 7y - 18 = 0, the next pivotal step is to factor the quadratic expression. Factoring involves breaking down the quadratic expression into a product of two binomials. This process hinges on finding two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. Specifically, we need to find two numbers that add up to the coefficient of the y term (which is 7) and multiply to the constant term (which is -18). This is a critical step because the correct numbers will allow us to rewrite the quadratic expression in a factored form, making it easier to find the solutions for y. To identify these numbers, we can systematically list the factors of -18 and check which pair adds up to 7. The factor pairs of -18 are: (1, -18), (-1, 18), (2, -9), (-2, 9), (3, -6), and (-3, 6). Among these pairs, the pair (-2, 9) satisfies both conditions: -2 + 9 = 7 and -2 * 9 = -18. Therefore, -2 and 9 are the numbers we need to factor the quadratic expression. Using these numbers, we can rewrite the quadratic expression as a product of two binomials: (y - 2)(y + 9). This factorization is the heart of the solving process, as it transforms the quadratic equation into a product of two linear factors. The factored form of the equation is (y - 2)(y + 9) = 0. This form is incredibly useful because it allows us to apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This principle forms the basis for finding the solutions for y. By successfully factoring the quadratic expression, we have simplified the problem into two simpler equations, each of which can be easily solved.
Step 3: Applying the Zero-Product Property
After successfully factoring the quadratic equation into the form (y - 2)(y + 9) = 0, the next crucial step is to apply the zero-product property. This property is a cornerstone of algebra and states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. In simpler terms, if A * B = 0, then either A = 0 or B = 0 (or both). This property allows us to break down the factored quadratic equation into two separate linear equations, each of which can be easily solved for y. Applying the zero-product property to (y - 2)(y + 9) = 0, we set each factor equal to zero, creating two equations: 1. y - 2 = 0 2. y + 9 = 0 Now, we have two simple linear equations to solve. To solve the first equation, y - 2 = 0, we add 2 to both sides, which isolates y and gives us y = 2. This is one of the solutions to the original quadratic equation. To solve the second equation, y + 9 = 0, we subtract 9 from both sides, which again isolates y and gives us y = -9. This is the second solution to the original quadratic equation. By applying the zero-product property, we have effectively transformed a single quadratic equation into two linear equations, each yielding a solution for y. These solutions, y = 2 and y = -9, are the values that, when substituted back into the original equation, will make the equation true. The zero-product property is a powerful tool in solving factored equations, not just quadratics, but any equation where the expression is a product of factors. It simplifies the problem by allowing us to focus on each factor individually, making it a cornerstone technique in algebra.
Step 4: Verifying the Solutions
Once we have obtained the solutions y = 2 and y = -9, it is crucial to verify these solutions by substituting them back into the original equation, y^2 + 7y = 18. This verification step is essential for ensuring that our solutions are correct and that no errors were made during the factoring or solving process. Substituting y = 2 into the original equation, we get: (2)^2 + 7(2) = 4 + 14 = 18. This confirms that y = 2 is indeed a solution, as it satisfies the equation. Next, we substitute y = -9 into the original equation: (-9)^2 + 7(-9) = 81 - 63 = 18. This also confirms that y = -9 is a valid solution, as it also satisfies the equation. By verifying both solutions, we can be confident that we have correctly solved the quadratic equation. This step is not just a formality; it is a critical part of the problem-solving process that helps to catch any potential errors. In more complex problems, errors can easily occur during the algebraic manipulations, and verifying the solutions is a reliable way to ensure accuracy. Moreover, the verification process reinforces the understanding of what it means to solve an equation. A solution is a value that makes the equation true, and substituting the solution back into the equation is a direct test of this property. This step solidifies the connection between the solutions and the original equation, enhancing comprehension of the problem. Therefore, always make it a practice to verify your solutions, especially in exams or when dealing with critical applications where accuracy is paramount. It is a small step that can save you from significant errors and boost your confidence in your problem-solving abilities.
Identifying the Correct Factored Form
Having successfully solved the equation y^2 + 7y = 18 by factoring, it is essential to connect the factoring process to the answer choices provided in a multiple-choice format. The factored form of the equation, which we derived as (y - 2)(y + 9) = 0, directly corresponds to one of the answer options. The options presented likely include variations of the factored form, and understanding how we arrived at our factored form allows us to identify the correct choice with confidence. In this case, the correct factored form is (y + 9)(y - 2) = 0, which is equivalent to (y - 2)(y + 9) = 0 due to the commutative property of multiplication. This means that the order in which the factors are written does not affect the equation's validity. Other options might include incorrect factorizations, such as (y + 2)(y + 9) = 0 or variations that do not accurately represent the original quadratic equation when expanded. It is crucial to recognize that the correct factored form, when expanded, should yield the standard form of the original equation, y^2 + 7y - 18 = 0. This serves as a quick way to verify that the chosen factored form is indeed the correct one. By expanding the factored form, we can check if it matches the original quadratic expression. For instance, expanding (y + 9)(y - 2) gives us y^2 - 2y + 9y - 18, which simplifies to y^2 + 7y - 18, confirming that it is the correct factorization. Understanding this connection between the factored form and the original equation is key to confidently selecting the right answer in a multiple-choice setting. It also reinforces the understanding of the factoring process and its relationship to the solutions of the quadratic equation. Therefore, when faced with multiple options, always consider the process of factoring and how it leads to the correct factored form, and verify your choice by expanding it back to the original equation.
Conclusion
In conclusion, solving the quadratic equation y^2 + 7y = 18 by factoring involves a series of essential steps, each building upon the previous one to arrive at the solutions. We began by understanding the concept of quadratic equations and their standard form, which is crucial for organizing the terms and identifying the coefficients. The first key step was rearranging the equation into the standard form, y^2 + 7y - 18 = 0, by subtracting 18 from both sides. This set the stage for the factoring process. Next, we focused on factoring the quadratic expression, which involved finding two numbers that add up to the coefficient of the y term (7) and multiply to the constant term (-18). This led us to the factors -2 and 9, allowing us to rewrite the equation in its factored form: (y - 2)(y + 9) = 0. The application of the zero-product property was the next critical step. By setting each factor equal to zero, we created two linear equations, y - 2 = 0 and y + 9 = 0, which were then easily solved to find the solutions y = 2 and y = -9. To ensure the accuracy of our solutions, we verified them by substituting them back into the original equation. This step confirmed that both y = 2 and y = -9 satisfy the equation, thus validating our factoring and solving process. Finally, we discussed how to identify the correct factored form among multiple choices, emphasizing the importance of understanding the factoring process and verifying the result by expanding the factored form back to the original quadratic expression. This comprehensive approach not only solves the specific equation but also provides a solid framework for tackling other quadratic equations by factoring. The ability to solve quadratic equations by factoring is a fundamental skill in algebra, and mastering these steps will undoubtedly enhance your mathematical proficiency and problem-solving abilities. Through practice and a clear understanding of the underlying principles, you can confidently approach and solve a wide range of quadratic equations.
