Solving (5y + 6)^2 = 24 A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of finding solutions for the quadratic equation (5y + 6)^2 = 24. Quadratic equations are fundamental in mathematics, appearing in various fields such as physics, engineering, and economics. Understanding how to solve them is crucial for problem-solving in these areas. This article aims to provide a step-by-step explanation of the solution process, ensuring a clear understanding for students and enthusiasts alike. Our primary focus will be on breaking down the equation, simplifying it, and ultimately arriving at the correct solutions. The solutions we are looking for are the values of 'y' that satisfy the given equation. We will explore the algebraic manipulations required to isolate 'y' and determine these values. This involves understanding the properties of square roots and how they interact with equations. We will also discuss the importance of verifying solutions to ensure their accuracy. This detailed approach will not only help in solving this specific equation but also equip you with the skills to tackle other quadratic equations confidently. Solving quadratic equations is a cornerstone of algebraic proficiency, and mastering it opens doors to more advanced mathematical concepts. So, let's embark on this journey together, demystifying the process and solidifying your understanding of quadratic equations.

Understanding Quadratic Equations

Before diving into the specific equation, let's first grasp the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The solutions to a quadratic equation are also known as its roots or zeros. These are the values of 'x' that make the equation true. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of equations. For instance, factoring is often the quickest method when the equation can be easily factored, while the quadratic formula is a more general method that can be applied to any quadratic equation. Understanding these methods and knowing when to apply them is key to solving quadratic equations efficiently. In our case, we will be using a combination of algebraic manipulation and the properties of square roots to solve the equation (5y + 6)^2 = 24. This approach will highlight the importance of simplifying the equation before applying specific methods. By breaking down the equation into smaller, manageable steps, we can clearly see the path to the solutions. This methodical approach is crucial for solving not just quadratic equations, but any mathematical problem.

Step-by-Step Solution of (5y + 6)^2 = 24

Now, let's tackle the given equation: (5y + 6)^2 = 24. The first step is to take the square root of both sides of the equation. This will help us eliminate the square on the left side and simplify the equation. Remember, when taking the square root, we need to consider both the positive and negative roots. This is because both the positive and negative square roots, when squared, will result in the same positive number. So, we have: √(5y + 6)^2 = ±√24. This simplifies to 5y + 6 = ±√24. Next, we can simplify √24. Since 24 can be factored as 4 * 6, we have √24 = √(4 * 6) = √4 * √6 = 2√6. So, our equation becomes: 5y + 6 = ±2√6. Now, we need to isolate 'y'. To do this, we first subtract 6 from both sides: 5y = -6 ± 2√6. Finally, we divide both sides by 5 to solve for 'y': y = (-6 ± 2√6) / 5. This gives us two possible solutions for 'y': y = (-6 + 2√6) / 5 and y = (-6 - 2√6) / 5. These are the two roots of the quadratic equation. It's important to note that these solutions are irrational numbers due to the presence of the square root of 6. Understanding how to manipulate and simplify expressions with square roots is crucial in solving quadratic equations and other algebraic problems. In the next section, we will verify these solutions to ensure their accuracy.

Verifying the Solutions

To ensure the accuracy of our solutions, we need to verify them by substituting them back into the original equation, (5y + 6)^2 = 24. Let's start with the first solution: y = (-6 + 2√6) / 5. Substitute this value of 'y' into the equation: (5((-6 + 2√6) / 5) + 6)^2 = 24. Simplifying inside the parentheses, we get: (-6 + 2√6 + 6)^2 = 24. This further simplifies to (2√6)^2 = 24. Squaring 2√6 gives us 4 * 6 = 24, which confirms that the first solution is correct. Now, let's verify the second solution: y = (-6 - 2√6) / 5. Substitute this value of 'y' into the equation: (5((-6 - 2√6) / 5) + 6)^2 = 24. Simplifying inside the parentheses, we get: (-6 - 2√6 + 6)^2 = 24. This simplifies to (-2√6)^2 = 24. Squaring -2√6 also gives us 4 * 6 = 24, confirming that the second solution is also correct. Verifying solutions is a crucial step in problem-solving. It helps to catch any errors made during the solution process and ensures that the answers obtained are indeed the correct ones. This practice is particularly important in quadratic equations, where the presence of square roots and multiple steps increases the chances of making a mistake. By verifying our solutions, we can be confident in our answers and the process we used to arrive at them.

The Solutions and Their Significance

Having solved the quadratic equation (5y + 6)^2 = 24 and verified the solutions, we have found that the values of 'y' that satisfy the equation are y = (-6 + 2√6) / 5 and y = (-6 - 2√6) / 5. These solutions are irrational numbers, which is not uncommon in quadratic equations. The presence of the square root of 6 in the solutions indicates that these values cannot be expressed as simple fractions. Understanding the nature of solutions is important in interpreting the results in real-world applications. For instance, if this equation represented a physical phenomenon, the solutions would provide specific values that satisfy the conditions of the problem. The fact that there are two solutions highlights a key characteristic of quadratic equations: they can have up to two distinct real roots. These roots represent the points where the quadratic function intersects the x-axis if we were to graph it. In some cases, quadratic equations may have only one real root (a repeated root) or no real roots (complex roots). The nature of the roots depends on the discriminant of the quadratic equation, which is a value derived from the coefficients of the equation. In summary, the solutions we have found are not just numerical answers; they are values that hold specific meaning within the context of the equation and its potential applications. Understanding this significance is crucial for applying mathematical concepts to solve real-world problems.

Conclusion

In conclusion, we have successfully solved the quadratic equation (5y + 6)^2 = 24 by systematically applying algebraic principles and the properties of square roots. We first simplified the equation by taking the square root of both sides, then isolated the variable 'y' to arrive at the solutions: y = (-6 + 2√6) / 5 and y = (-6 - 2√6) / 5. We further verified these solutions by substituting them back into the original equation, confirming their accuracy. This process underscores the importance of a methodical approach in solving mathematical problems. Breaking down complex equations into smaller, manageable steps allows for clarity and reduces the likelihood of errors. The solutions we obtained are irrational numbers, highlighting the diverse nature of solutions that can arise from quadratic equations. This exercise not only provides the specific solutions to this equation but also reinforces the fundamental concepts of quadratic equations and algebraic manipulation. The ability to solve quadratic equations is a valuable skill in mathematics and its applications. By mastering these techniques, you can confidently tackle a wide range of problems in various fields. Remember, practice is key to proficiency, so continue to explore and solve different types of quadratic equations to further enhance your understanding and skills. The journey of mathematical learning is continuous, and each problem solved adds to your knowledge and confidence.

The correct answer is A: y = (-6 + 2√6) / 5 and y = (-6 - 2√6) / 5