Solving Quadratic Equations Step By Step A Guide To 3(x-4)^2 = 75

In the realm of mathematics, quadratic equations hold a prominent position, representing a fundamental concept with wide-ranging applications. These equations, characterized by their second-degree polynomial form, often pose a challenge to students and enthusiasts alike. This comprehensive guide aims to demystify the process of solving quadratic equations, providing a step-by-step approach that empowers you to tackle even the most complex problems. We will delve into the intricacies of solving the specific quadratic equation 3(x-4)^2 = 75, showcasing the techniques and strategies involved in arriving at the correct solutions. Understanding quadratic equations is crucial not only for academic success but also for various real-world applications, from physics and engineering to finance and economics. Mastering the methods to solve them opens doors to a deeper understanding of mathematical principles and their practical implications.

Demystifying the Quadratic Equation: 3(x-4)^2 = 75

Our focus today is on unraveling the solutions to the quadratic equation 3(x-4)^2 = 75. This particular equation presents a slightly different format than the standard quadratic form, requiring a strategic approach to isolate the variable 'x' and determine its possible values. Solving this equation involves a combination of algebraic manipulations, including simplifying, taking square roots, and isolating the variable. By carefully executing each step, we can arrive at the correct solutions and gain a deeper understanding of the underlying principles of quadratic equations. The process of solving this equation not only provides the specific answers but also reinforces the general techniques applicable to a wider range of quadratic problems. This hands-on approach fosters a deeper conceptual understanding and enhances problem-solving skills.

Step 1: Isolating the Squared Term

The initial step in solving the quadratic equation 3(x-4)^2 = 75 involves isolating the squared term, (x-4)^2. This is achieved by dividing both sides of the equation by the coefficient 3. This operation maintains the equality of the equation while simplifying the expression, paving the way for subsequent steps. By isolating the squared term, we effectively remove the influence of the coefficient and focus on the core quadratic expression. This simplification is crucial for applying further algebraic manipulations and ultimately solving for the unknown variable 'x'. The resulting equation, (x-4)^2 = 25, is now in a more manageable form, allowing us to proceed with taking the square root of both sides.

Step 2: Taking the Square Root of Both Sides

With the squared term isolated, the next logical step is to take the square root of both sides of the equation (x-4)^2 = 25. This operation is crucial for eliminating the square and bringing us closer to isolating 'x'. However, it's essential to remember that taking the square root introduces two possible solutions: a positive and a negative root. This is because both the positive and negative square roots, when squared, will result in the same positive value. In this case, the square root of 25 is both +5 and -5. Therefore, we obtain two separate equations: x-4 = 5 and x-4 = -5. This recognition of both positive and negative roots is a fundamental aspect of solving quadratic equations and ensures that we capture all possible solutions.

Step 3: Solving for x

Now, we have two simple linear equations: x-4 = 5 and x-4 = -5. Solving for 'x' in each equation involves adding 4 to both sides. For the first equation, x-4 = 5, adding 4 to both sides gives us x = 9. For the second equation, x-4 = -5, adding 4 to both sides results in x = -1. These two values, x = 9 and x = -1, represent the solutions to the original quadratic equation. They are the values that, when substituted back into the original equation, will satisfy the equality. This step highlights the importance of basic algebraic manipulation in isolating the variable and arriving at the final solutions.

Step 4: Verifying the Solutions

To ensure the accuracy of our solutions, it's always a good practice to verify them by substituting them back into the original equation. Let's first substitute x = 9 into the equation 3(x-4)^2 = 75. We get 3(9-4)^2 = 3(5)^2 = 3(25) = 75, which confirms that x = 9 is a valid solution. Next, let's substitute x = -1 into the equation. We get 3(-1-4)^2 = 3(-5)^2 = 3(25) = 75, which also confirms that x = -1 is a valid solution. This verification step provides confidence in our solutions and reinforces the understanding of how the solutions relate to the original equation. It also helps to identify any potential errors in the solving process.

The Answer

Therefore, the solutions to the quadratic equation 3(x-4)^2 = 75 are x = -1 and x = 9, which corresponds to option D. This journey through the steps of solving this quadratic equation highlights the importance of methodical algebraic manipulation, attention to detail, and the understanding of fundamental mathematical principles. By mastering these techniques, you can confidently tackle a wide range of quadratic equations and further your mathematical prowess.

Why Other Options are Incorrect

Understanding why the other options are incorrect is just as crucial as knowing the correct answer. Let's analyze why options A, B, and C are not the solutions to the equation 3(x-4)^2 = 75. This process will reinforce our understanding of quadratic equations and the importance of accurate problem-solving.

• Option A: x = -9 and x = 1 If we substitute x = -9 into the equation, we get 3(-9-4)^2 = 3(-13)^2 = 3(169) = 507, which is not equal to 75. Similarly, if we substitute x = 1, we get 3(1-4)^2 = 3(-3)^2 = 3(9) = 27, which is also not equal to 75. Therefore, option A is incorrect.
• Option B: x = -5 and x = 5 Substituting x = -5, we get 3(-5-4)^2 = 3(-9)^2 = 3(81) = 243, which is not equal to 75. Substituting x = 5, we get 3(5-4)^2 = 3(1)^2 = 3(1) = 3, which is also not equal to 75. Hence, option B is incorrect.
• Option C: x = -4 and x = 4 If we substitute x = -4, we get 3(-4-4)^2 = 3(-8)^2 = 3(64) = 192, which is not equal to 75. Substituting x = 4, we get 3(4-4)^2 = 3(0)^2 = 3(0) = 0, which is also not equal to 75. Consequently, option C is incorrect.

These analyses demonstrate the importance of verifying solutions and understanding the impact of each step in the solving process. By carefully substituting the potential solutions back into the original equation, we can confidently identify the correct answers and avoid common errors.

Mastering Quadratic Equations: Key Takeaways

Solving quadratic equations is a fundamental skill in mathematics, and mastering it requires a combination of understanding the underlying concepts and practicing problem-solving techniques. Here are some key takeaways to help you excel in this area:

1. Understand the Standard Form: Familiarize yourself with the standard form of a quadratic equation, which is ax^2 + bx + c = 0. Recognizing this form will help you identify quadratic equations and apply appropriate solving methods.
2. Isolate the Squared Term: When dealing with equations in the form of (x-h)^2 = k, the first step is to isolate the squared term. This simplifies the equation and allows you to proceed with taking the square root.
3. Remember the Positive and Negative Roots: When taking the square root of both sides of an equation, always remember to consider both the positive and negative roots. This is crucial for capturing all possible solutions.
4. Verify Your Solutions: Always verify your solutions by substituting them back into the original equation. This ensures accuracy and helps you identify any potential errors.
5. Practice Regularly: The key to mastering quadratic equations is consistent practice. Solve a variety of problems to develop your skills and build confidence.
6. Explore Different Methods: While this guide focused on solving by isolating the squared term, there are other methods such as factoring, completing the square, and using the quadratic formula. Explore these methods to broaden your problem-solving toolkit.

By keeping these takeaways in mind and practicing diligently, you can develop a strong understanding of quadratic equations and confidently solve a wide range of problems.

Conclusion: Embracing the Power of Quadratic Equations

In conclusion, solving quadratic equations is a vital skill in mathematics with far-reaching applications. This guide has provided a comprehensive walkthrough of solving the specific equation 3(x-4)^2 = 75, highlighting the key steps and concepts involved. By understanding the process of isolating the squared term, taking square roots, and verifying solutions, you can confidently tackle similar problems and expand your mathematical abilities. Remember, practice is the key to mastery. The more you engage with quadratic equations, the more proficient you will become in solving them. Embrace the challenge, explore different methods, and unlock the power of quadratic equations in your mathematical journey. The solutions to quadratic equations are not just numbers; they are keys to understanding and solving problems in various fields, from science and engineering to economics and finance. So, continue to explore, learn, and grow in your mathematical endeavors.