Solving (y+8)^2-63=0 A Step-by-Step Guide

Introduction: Delving into the Realm of Quadratic Equations

In the captivating world of mathematics, quadratic equations stand as fundamental pillars, weaving their way through various disciplines and real-world applications. At its core, a quadratic equation is a polynomial equation of the second degree, characterized by the presence of a squared term. These equations often appear in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable we seek to unravel. In this comprehensive guide, we embark on a journey to solve the equation (y+8)^2-63=0, where y is a real number. Our exploration will involve simplifying the equation, employing algebraic techniques, and ultimately arriving at the solution(s), meticulously simplified and presented in a clear and concise manner. The significance of mastering quadratic equations extends far beyond academic exercises, permeating fields such as physics, engineering, economics, and computer science. From modeling projectile motion to optimizing financial investments, quadratic equations serve as indispensable tools for understanding and solving complex problems. Therefore, a solid grasp of quadratic equation solving techniques empowers individuals to navigate the intricacies of the world around them, fostering critical thinking and problem-solving skills that are highly valued across various domains.

Problem Statement: Deconstructing the Equation (y+8)^2-63=0

At the heart of our mathematical endeavor lies the equation (y+8)^2-63=0. This equation, while seemingly simple in its structure, holds a gateway to a deeper understanding of quadratic equations and their solutions. Our primary objective is to determine the real number values of y that satisfy this equation. In other words, we seek the values of y that, when substituted into the equation, render the left-hand side equal to zero. Before we delve into the solution process, let's dissect the equation to gain a clearer perspective. The equation features a squared term, (y+8)^2, which indicates that it is indeed a quadratic equation. The constant term, -63, adds another layer of complexity, requiring us to employ strategic algebraic manipulations to isolate the variable y. The expression (y+8) within the parentheses suggests a shift in the variable, which we must carefully consider during our solution process. By understanding the individual components of the equation, we can formulate a plan of action to solve for y effectively. This initial analysis serves as a crucial foundation for our subsequent steps, guiding us towards the correct solution(s).

Solution Strategy: A Step-by-Step Approach

To conquer the equation (y+8)^2-63=0, we employ a systematic approach that breaks down the problem into manageable steps. Our strategy involves isolating the squared term, taking the square root of both sides, and then solving for y. This method leverages the fundamental properties of algebraic equations and the inverse relationship between squaring and taking the square root. Step 1: Isolate the Squared Term Our initial focus is to isolate the (y+8)^2 term on one side of the equation. To achieve this, we add 63 to both sides of the equation. This operation maintains the equality and simplifies the equation by eliminating the constant term on the left-hand side. The resulting equation becomes (y+8)^2 = 63. Step 2: Take the Square Root of Both Sides Now that the squared term is isolated, we can take the square root of both sides of the equation. This step is crucial as it undoes the squaring operation and allows us to work directly with the expression (y+8). However, it's essential to remember that taking the square root introduces both positive and negative solutions. The square root of 63 can be expressed as ±√63. Therefore, we have y+8 = ±√63. Step 3: Simplify the Radical Before we proceed further, let's simplify the radical √63. We can factor 63 as 9 × 7, where 9 is a perfect square. Therefore, √63 can be written as √(9 × 7) = √9 × √7 = 3√7. This simplification makes the subsequent calculations easier. Our equation now becomes y+8 = ±3√7. Step 4: Solve for y The final step involves isolating y by subtracting 8 from both sides of the equation. This operation yields the solutions for y. We have y = -8 ± 3√7. This expression represents two distinct solutions: y = -8 + 3√7 and y = -8 - 3√7. These solutions are the real number values of y that satisfy the original equation.

Detailed Solution: Unraveling the Mathematical Steps

To provide a comprehensive understanding of the solution process, let's delve into a step-by-step breakdown, meticulously illustrating each algebraic manipulation. Step 1: Isolate the Squared Term Starting with the equation (y+8)^2-63=0, we add 63 to both sides: (y+8)^2-63+63=0+63 This simplifies to: (y+8)^2=63 Step 2: Take the Square Root of Both Sides Taking the square root of both sides, we get: √(y+8)^2 = ±√63 Remember that we must consider both positive and negative roots. This gives us: y+8 = ±√63 Step 3: Simplify the Radical We simplify √63 by factoring 63 as 9 × 7: √63 = √(9 × 7) = √9 × √7 = 3√7 Substituting this back into the equation, we have: y+8 = ±3√7 Step 4: Solve for y To isolate y, we subtract 8 from both sides: y+8-8 = -8 ± 3√7 This simplifies to: y = -8 ± 3√7 Therefore, the two solutions for y are: y = -8 + 3√7 and y = -8 - 3√7 These are the exact solutions to the equation, expressed in simplified radical form. By meticulously outlining each step, we gain a deeper appreciation for the mathematical process involved in solving the equation.

Presenting the Solution: Clarity and Conciseness

Having meticulously worked through the solution process, we now present the solutions in a clear and concise manner. The equation (y+8)^2-63=0 has two distinct real number solutions. These solutions are: y = -8 + 3√7 y = -8 - 3√7 To enhance clarity and readability, we can express these solutions as a comma-separated list: y = -8 + 3√7, -8 - 3√7 This format adheres to the prompt's instructions, ensuring that the solutions are presented in a user-friendly manner. The solutions are expressed in simplified radical form, which is the most accurate and precise way to represent them. Decimal approximations can be used for practical applications, but the exact solutions provide a deeper understanding of the mathematical nature of the problem. By presenting the solutions in this manner, we effectively communicate the results of our mathematical endeavor, allowing others to readily grasp the answers and their significance.

Alternative Solution Methods: Expanding Our Mathematical Toolkit

While we have successfully solved the equation (y+8)^2-63=0 using the square root method, it's insightful to explore alternative solution methods. This not only broadens our mathematical toolkit but also provides a deeper understanding of the equation's structure. Method 1: Expanding and Factoring We can expand the squared term in the equation and rewrite it in the standard quadratic form ax^2 + bx + c = 0. Then, we can attempt to factor the quadratic expression. Expanding (y+8)^2, we get: (y+8)^2 = y^2 + 16y + 64 Substituting this back into the original equation, we have: y^2 + 16y + 64 - 63 = 0 Simplifying, we get: y^2 + 16y + 1 = 0 Now, we attempt to factor this quadratic expression. However, it's not readily factorable using integers. Therefore, this method might not be the most efficient in this case. Method 2: Quadratic Formula The quadratic formula provides a general solution for any quadratic equation in the form ax^2 + bx + c = 0. The formula is: y = (-b ± √(b^2 - 4ac)) / (2a) In our case, the equation y^2 + 16y + 1 = 0 has coefficients a = 1, b = 16, and c = 1. Substituting these values into the quadratic formula, we get: y = (-16 ± √(16^2 - 4 × 1 × 1)) / (2 × 1) Simplifying, we have: y = (-16 ± √(256 - 4)) / 2 y = (-16 ± √252) / 2 We can simplify √252 by factoring it as √(36 × 7) = 6√7. Substituting this back into the equation, we get: y = (-16 ± 6√7) / 2 Dividing both terms in the numerator by 2, we have: y = -8 ± 3√7 This yields the same solutions we obtained using the square root method: y = -8 + 3√7 and y = -8 - 3√7. By exploring these alternative methods, we reinforce our understanding of quadratic equations and their solutions. Each method offers a unique perspective and highlights the versatility of algebraic techniques.

Conclusion: Mastering Quadratic Equations

In this comprehensive guide, we have successfully navigated the solution of the equation (y+8)^2-63=0, where y is a real number. Our journey has involved simplifying the equation, employing algebraic techniques, and ultimately arriving at the solutions: y = -8 + 3√7 and y = -8 - 3√7. We presented these solutions in a clear and concise manner, adhering to the prompt's instructions. Furthermore, we explored alternative solution methods, such as expanding and factoring and the quadratic formula, demonstrating the breadth of our mathematical toolkit. Mastering quadratic equations is a fundamental step in mathematical proficiency. These equations serve as building blocks for more advanced concepts and find applications in various fields. By understanding the underlying principles and techniques for solving quadratic equations, we empower ourselves to tackle complex problems and make informed decisions. This guide has provided a thorough exploration of the solution process, equipping readers with the knowledge and skills to confidently approach quadratic equations and their applications. The ability to solve quadratic equations is not merely an academic exercise; it is a valuable asset that enhances critical thinking and problem-solving abilities, fostering success in diverse endeavors.