Solve (w+7)^2 = 18: Finding Real Solutions Step-by-Step
Introduction
In this comprehensive guide, we will walk you through the process of finding all real solutions to the equation (w+7)^2 = 18. This type of equation, a quadratic equation in disguise, often appears in algebra and calculus problems. Understanding how to solve it is crucial for building a strong foundation in mathematics. We will break down the problem into manageable steps, explaining the underlying concepts and techniques along the way. Whether you're a student looking to improve your algebra skills or simply someone who enjoys mathematical puzzles, this guide will provide you with the knowledge and tools to tackle similar problems with confidence. Let's embark on this mathematical journey together and uncover the real solutions to the equation. This article aims to provide a clear, concise, and comprehensive explanation, ensuring that you grasp the fundamental principles involved in solving quadratic equations of this form.
Understanding the Equation
Before we dive into the solution, let's take a closer look at the equation (w+7)^2 = 18. This equation represents a quadratic relationship between the variable 'w' and a constant. The left-hand side, (w+7)^2, is a perfect square, which means it can be expanded as (w+7)(w+7). The right-hand side, 18, is a constant value. Our goal is to find all the values of 'w' that satisfy this equation. To do this, we'll need to use algebraic techniques to isolate 'w' and determine its possible values. The equation's structure suggests that taking the square root of both sides will be a key step in our solution process. Recognizing the form of the equation and understanding its components is the first step towards finding the solutions. We will also discuss why this equation can have multiple solutions, a characteristic feature of quadratic equations. The square on the left side is a crucial element that dictates the method we will employ to solve for w. In essence, we are seeking the values of w that, when 7 is added and the result is squared, yield 18.
Step-by-Step Solution
Now, let's proceed with solving the equation (w+7)^2 = 18 step-by-step:
Step 1: Take the Square Root of Both Sides
The first step in solving this equation is to eliminate the square on the left-hand side. We can do this by taking the square root of both sides of the equation. Remember that when we take the square root, we need to consider both the positive and negative roots. This is because both a positive number squared and a negative number squared will result in a positive number. So, taking the square root of both sides of (w+7)^2 = 18 gives us:
√(w+7)^2 = ±√18
w + 7 = ±√18
It is crucial to remember the ± sign, as this accounts for both possible solutions. Neglecting this can lead to missing one of the roots. The square root operation is the inverse of squaring, effectively undoing the (w+7)^2 term. The introduction of the ± sign reflects the fundamental property that both positive and negative values, when squared, can produce the same positive result. This step is a cornerstone of solving equations involving squared terms. The importance of considering both positive and negative roots cannot be overstated, as it ensures that we capture the complete set of solutions.
Step 2: Simplify the Square Root
Next, we simplify the square root of 18. We can express 18 as the product of its prime factors: 18 = 2 × 3 × 3 = 2 × 3². This allows us to simplify the square root as follows:
√18 = √(2 × 3²) = √2 × √3² = 3√2
So, our equation now becomes:
w + 7 = ±3√2
Simplifying the square root makes the equation easier to work with and allows us to express the solutions in their simplest form. This simplification relies on the properties of square roots, particularly the rule that √(a × b) = √a × √b. Recognizing perfect square factors within the square root is key to this simplification process. In this case, identifying 3² as a factor of 18 allows us to extract it from the square root, leaving the simplified term 3√2. This step not only makes the equation more manageable but also adheres to the standard practice of expressing solutions in their most simplified radical form. The ability to efficiently simplify square roots is a valuable skill in algebra and beyond.
Step 3: Isolate 'w'
To isolate 'w', we subtract 7 from both sides of the equation:
w = -7 ± 3√2
This gives us two possible solutions for 'w':
w = -7 + 3√2 and w = -7 - 3√2
Isolating the variable is a fundamental algebraic technique used to solve for unknown values. In this case, subtracting 7 from both sides effectively moves the constant term to the right side of the equation, leaving 'w' by itself on the left side. This step directly reveals the two possible values of 'w' that satisfy the original equation. The plus-minus (±) sign elegantly encapsulates both solutions in a single expression. This step is a direct application of the properties of equality, ensuring that the equation remains balanced while we manipulate it to solve for the unknown. The resulting expression, w = -7 ± 3√2, represents the complete solution set for the equation.
Final Solutions
Therefore, the real solutions to the equation (w+7)^2 = 18 are:
w = -7 + 3√2, -7 - 3√2
These are the two values of 'w' that, when substituted back into the original equation, will make the equation true. We have successfully found all real solutions by systematically applying algebraic techniques. This final step presents the solutions in a clear and concise manner, fulfilling the objective of the problem. The solutions are expressed in their simplified radical form, which is the conventional way to represent such values. The separation of the two solutions by a comma emphasizes that we have identified two distinct real values that satisfy the equation. The process we followed demonstrates a rigorous and systematic approach to solving quadratic equations of this type. These solutions represent the culmination of our step-by-step process, providing the answer to the original problem.
Verification (Optional)
To verify our solutions, we can substitute each value of 'w' back into the original equation and check if it holds true. Let's start with w = -7 + 3√2:
((-7 + 3√2) + 7)² = (3√2)² = 9 × 2 = 18
Now, let's check w = -7 - 3√2:
((-7 - 3√2) + 7)² = (-3√2)² = 9 × 2 = 18
Both solutions satisfy the original equation, confirming our results.
Verifying the solutions is an important step in the problem-solving process, as it ensures that our calculations are accurate and that the solutions we obtained are indeed correct. By substituting the solutions back into the original equation, we can confirm whether they make the equation true. This step adds a layer of confidence and completeness to our solution. The verification process reinforces the understanding of the equation and the meaning of its solutions. It also demonstrates the interconnectedness of the algebraic steps we took to arrive at the answer. This optional step is highly recommended, especially in more complex problems, to avoid errors and ensure the validity of the results.
Conclusion
In this guide, we have successfully found all real solutions to the equation (w+7)^2 = 18. We started by understanding the equation and recognizing its structure. Then, we followed a step-by-step solution process, which included taking the square root of both sides, simplifying the square root, and isolating the variable 'w'. We obtained two real solutions: w = -7 + 3√2 and w = -7 - 3√2. We also verified our solutions by substituting them back into the original equation. This process demonstrates the power of algebraic techniques in solving equations and finding unknown values. Understanding these techniques is essential for success in mathematics and related fields. We hope this guide has been helpful and that you now have a clearer understanding of how to solve equations of this type. Remember to practice these techniques on other similar problems to solidify your understanding and build your problem-solving skills. The ability to solve equations like this is a valuable asset in mathematics, science, and engineering. This conclusion summarizes the journey we undertook, highlighting the key steps and the final results. It also emphasizes the importance of understanding the underlying concepts and practicing the techniques. The successful solution of this equation showcases the effectiveness of the step-by-step approach and the power of algebraic manipulation. We encourage readers to apply these skills to other problems and to continue their exploration of mathematics.
