Solving (x-5)^2 = 9 A Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, quadratic equations hold a significant place. These equations, characterized by the presence of a squared term, often appear in diverse fields like physics, engineering, and economics. In this comprehensive guide, we will delve into the process of solving the specific quadratic equation (x-5)^2 = 9, providing a step-by-step approach that will empower you to tackle similar problems with confidence. This guide aims to enhance your understanding of quadratic equations and equip you with the necessary tools to solve them effectively. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this guide will provide a clear and concise approach to solving quadratic equations.

Understanding Quadratic Equations: Before we jump into solving the equation, let's take a moment to understand what quadratic equations are. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable in the equation 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. Quadratic equations can have two, one, or no real solutions, depending on the values of the coefficients. The solutions to a quadratic equation are also known as the roots of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The choice of method depends on the specific equation and the ease with which it can be applied. In this guide, we will focus on solving the equation (x-5)^2 = 9, which is a special case that can be solved using the square root property. This method involves isolating the squared term and then taking the square root of both sides of the equation. Understanding the structure of quadratic equations and the various methods to solve them is crucial for mastering algebra and its applications in various fields.

The Square Root Property: The square root property is a powerful tool for solving quadratic equations that are in the form of a squared term equal to a constant. This property states that if x^2 = k, where k is a non-negative constant, then x = ±√k. In other words, if the square of a variable is equal to a constant, then the variable is equal to the positive or negative square root of the constant. The square root property is particularly useful when the quadratic equation can be easily rearranged into the form x^2 = k. This method simplifies the process of finding the solutions by directly addressing the squared term. It avoids the need for factoring or using the quadratic formula, which can be more complex and time-consuming. To apply the square root property, it is essential to first isolate the squared term on one side of the equation. Once the squared term is isolated, taking the square root of both sides will yield the solutions. Remember to consider both the positive and negative square roots, as both values will satisfy the equation. The square root property is a fundamental concept in algebra and is widely used in solving various types of equations, making it an essential tool for any mathematics student or professional.

Step-by-Step Solution of (x-5)^2 = 9

Now, let's apply our knowledge to solve the equation (x-5)^2 = 9. We'll break down the process into clear, manageable steps.

Step 1: Taking the Square Root of Both Sides

The first critical step in solving the equation (x-5)^2 = 9 is to apply the square root property. This involves taking the square root of both sides of the equation. By doing this, we aim to eliminate the square on the left side, which will help us isolate the variable x. When we take the square root of a squared term, we effectively undo the squaring operation. However, it's crucial to remember that when taking the square root of a constant, we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will yield the same positive result. For example, both 3 and -3, when squared, result in 9. Therefore, when we take the square root of 9, we must consider both +3 and -3. This step is fundamental to ensuring we find all possible solutions to the equation. Failing to consider both roots would lead to an incomplete solution set. Therefore, taking the square root of both sides is the cornerstone of this solving process. Applying this to our equation, we get √(x-5)^2 = ±√9. The left side simplifies to x-5, and the right side becomes ±3.

Step 2: Isolating x

After taking the square root of both sides, our equation now looks like x - 5 = ±3. The next step is to isolate the variable x. This means we need to get x by itself on one side of the equation. To do this, we perform the inverse operation of what is being done to x. In this case, 5 is being subtracted from x. The inverse operation of subtraction is addition. Therefore, we add 5 to both sides of the equation. This ensures that the equation remains balanced, maintaining the equality. Adding 5 to both sides of x - 5 = ±3 gives us x - 5 + 5 = ±3 + 5. Simplifying this, we get x = ±3 + 5. This step is crucial because it separates x from the other terms in the equation, bringing us closer to the solution. By adding 5 to both sides, we effectively cancel out the -5 on the left side, leaving x isolated. This process is a fundamental algebraic technique used in solving various types of equations. It's important to perform the same operation on both sides of the equation to maintain balance and ensure the solution remains valid.

Step 3: Finding the Two Possible Solutions

Now that we have isolated x, we have the equation x = ±3 + 5. This equation represents two distinct possibilities for the value of x, due to the ± sign. We must consider both the positive and negative cases to find all possible solutions. First, let's consider the positive case: x = +3 + 5. Adding 3 and 5 gives us x = 8. So, one possible solution is x = 8. Next, we consider the negative case: x = -3 + 5. Adding -3 and 5 gives us x = 2. Therefore, the second possible solution is x = 2. These two solutions, x = 8 and x = 2, are the values that will satisfy the original equation (x-5)^2 = 9. To verify these solutions, we can substitute each value back into the original equation and check if the equation holds true. This step is crucial to ensure that the solutions we found are correct. In summary, by considering both the positive and negative cases, we have successfully found the two possible solutions for x.

Verifying the Solutions

To ensure the accuracy of our solutions, it is essential to verify them by substituting them back into the original equation. This process confirms that the values we found for x indeed satisfy the equation (x-5)^2 = 9. Let's verify each solution individually.

Verifying x = 8

To verify the solution x = 8, we substitute this value into the original equation (x-5)^2 = 9. Replacing x with 8, we get (8-5)^2 = 9. First, we simplify the expression inside the parentheses: 8 - 5 = 3. So, the equation becomes 3^2 = 9. Now, we evaluate the square: 3^2 = 3 * 3 = 9. Therefore, we have 9 = 9, which is a true statement. This confirms that x = 8 is indeed a valid solution to the equation (x-5)^2 = 9. Verification is a crucial step in the problem-solving process, as it helps to identify any potential errors made during the solution process. By substituting the solution back into the original equation, we can ensure that the solution is correct and satisfies the given conditions. This step provides confidence in the accuracy of the solution and reinforces the understanding of the equation.

Verifying x = 2

Now, let's verify the second solution, x = 2. We follow the same process as before, substituting x = 2 into the original equation (x-5)^2 = 9. Replacing x with 2, we get (2-5)^2 = 9. First, we simplify the expression inside the parentheses: 2 - 5 = -3. So, the equation becomes (-3)^2 = 9. Now, we evaluate the square: (-3)^2 = (-3) * (-3) = 9. Therefore, we have 9 = 9, which is a true statement. This confirms that x = 2 is also a valid solution to the equation (x-5)^2 = 9. The verification process is a fundamental aspect of mathematical problem-solving. It ensures that the solutions obtained are accurate and satisfy the given equation. By verifying each solution, we gain confidence in the correctness of our work and reinforce our understanding of the concepts involved. In this case, both x = 8 and x = 2 satisfy the original equation, demonstrating that our step-by-step solution was accurate and complete.

Conclusion

In this comprehensive guide, we have successfully solved the quadratic equation (x-5)^2 = 9. We began by understanding the basics of quadratic equations and the square root property, which is a key tool for solving equations in this form. Then, we systematically applied the steps to solve the equation: taking the square root of both sides, isolating x, and finding the two possible solutions. Finally, we verified our solutions to ensure their accuracy. The solutions to the equation (x-5)^2 = 9 are x = 8 and x = 2. This process not only provides the solutions to this specific equation but also equips you with the knowledge and skills to solve similar quadratic equations. Understanding and applying these steps will enhance your problem-solving abilities in mathematics and related fields. Remember, practice is key to mastering these concepts. The more you solve quadratic equations, the more confident and proficient you will become. This guide serves as a solid foundation for further exploration of algebraic concepts and techniques. Keep practicing and exploring, and you will find yourself becoming more adept at solving mathematical problems.

  • Solving quadratic equations
  • Square root property
  • Step-by-step solution
  • Verifying solutions
  • Algebraic equations
  • Mathematics guide
  • Equation solving
  • Quadratic equation solutions
  • Isolate the variable
  • Positive and negative roots