Solving $x^2 - 1 = 399$ A Step-by-Step Guide

Quadratic equations are fundamental in algebra, and mastering their solutions is crucial for various mathematical and real-world applications. Today, we will walk through the step-by-step process of solving the quadratic equation $x^2 - 1 = 399$. This exercise will not only provide a solution but also enhance your understanding of the methods used to tackle such problems. We'll break down each step, ensuring clarity and building a solid foundation for future problem-solving. Understanding how to solve quadratic equations opens doors to more complex mathematical concepts and their applications in physics, engineering, and economics. By the end of this article, you'll be equipped with the knowledge to confidently solve similar equations and appreciate the elegance of algebraic solutions. Our approach will be detailed and easy to follow, suitable for learners at various levels of mathematical proficiency. Solving quadratic equations involves identifying the coefficients, understanding the standard form, and applying appropriate algebraic techniques. Whether you're a student preparing for an exam or simply seeking to refresh your algebra skills, this guide will serve as a valuable resource. We'll also touch on common pitfalls to avoid, ensuring that you develop a robust and accurate problem-solving methodology. Let's dive into the specifics of our example equation and unlock its solution.

Understanding the Problem

First, let's restate the problem clearly. We are tasked with finding the solutions to the equation $x^2 - 1 = 399$. This equation is a classic example of a quadratic equation, which generally takes the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Before jumping into calculations, it's essential to understand what a solution to this equation actually means. A solution, or root, is a value of $x$ that, when substituted into the equation, makes the equation true. In other words, we're looking for values of $x$ that satisfy the given condition. Recognizing the structure of the equation as a quadratic form helps us determine the best approach for solving it. In this case, we notice that the equation is relatively simple, lacking a linear term (the term with $x$ to the first power). This suggests that we can isolate $x^2$ and then take the square root to find the solutions. However, we must be cautious and remember that taking the square root can yield both positive and negative solutions. This initial understanding sets the stage for a methodical solution process, ensuring that we consider all possibilities and arrive at the correct answer. Identifying the type of equation we're dealing with is the first crucial step in any mathematical problem-solving endeavor.

Step 1: Isolate $x^2$

To begin solving the equation $x^2 - 1 = 399$, our first step is to isolate the term containing $x^2$. This means we want to get $x^2$ by itself on one side of the equation. To do this, we perform the inverse operation of subtracting 1, which is adding 1, to both sides of the equation. This maintains the balance of the equation and allows us to simplify it. So, we add 1 to both sides:

$x^2 - 1 + 1 = 399 + 1$

Simplifying this, we get:

$x^2 = 400$

Now, we have successfully isolated $x^2$. This step is crucial because it sets us up to use the square root operation to find the values of $x$. Isolating the variable term is a common technique in solving algebraic equations and is a fundamental skill to develop. By isolating $x^2$, we've transformed the equation into a simpler form that is more amenable to finding the solutions. This process highlights the importance of performing the same operations on both sides of an equation to maintain equality, a core principle of algebraic manipulation. With $x^2$ isolated, we are now ready to move on to the next step, which involves taking the square root of both sides to find the possible values of $x$. This methodical approach ensures accuracy and minimizes the chances of error.

Step 2: Take the Square Root

Now that we have the equation $x^2 = 400$, the next step is to find the values of $x$ that satisfy this equation. To do this, we take the square root of both sides. It's crucial to remember that when we take the square root of a number, we must consider both the positive and negative roots. This is because both the positive and negative values, when squared, will result in the same positive number. So, we have:

$\sqrt{x^2} = \pm\sqrt{400}$

The square root of $x^2$ is simply $x$, and the square root of 400 is 20. Therefore, we have:

$x = \pm 20$

This means that $x$ can be either 20 or -20. Both of these values, when squared, will equal 400. This step underscores the importance of considering both positive and negative solutions when dealing with square roots in equations. Failing to do so can lead to missing one of the correct answers. Taking the square root is a powerful operation for solving equations where the variable is squared, but it requires careful attention to the signs. By considering both possibilities, we ensure that we have found all the solutions to the equation. This careful approach is essential for accuracy in algebraic problem-solving. We have now found the two possible values for $x$, which brings us to the final step of verifying our solutions.

Step 3: Verify the Solutions

After finding the possible solutions, it's always a good practice to verify them. This step ensures that the values we found actually satisfy the original equation. We found that $x = 20$ and $x = -20$ are potential solutions to the equation $x^2 - 1 = 399$. Let's substitute each value back into the original equation to check:

For $x = 20$:

$(20)^2 - 1 = 400 - 1 = 399$

This is true, so $x = 20$ is indeed a solution.

For $x = -20$:

$(-20)^2 - 1 = 400 - 1 = 399$

This is also true, so $x = -20$ is a solution as well.

By verifying both solutions, we can be confident that our answers are correct. This step is particularly important in exams or assessments, where a simple check can prevent careless errors. Verification reinforces the understanding of what it means for a value to be a solution to an equation. It also highlights the symmetry in the solutions that arises from squaring a number – both the positive and negative values yield the same result. This methodical approach to problem-solving, including the crucial step of verification, is a hallmark of mathematical rigor and accuracy. With both solutions verified, we can confidently state our final answer.

Final Answer

After walking through the steps and verifying our solutions, we can confidently state the final answer. The solutions to the equation $x^2 - 1 = 399$ are:

$x = 20$ and $x = -20$

Therefore, the correct answer is option A. This comprehensive solution demonstrates the process of solving a quadratic equation, emphasizing the importance of isolating the variable, considering both positive and negative roots, and verifying the solutions. This methodical approach is applicable to a wide range of algebraic problems and is a valuable skill for anyone studying mathematics. Understanding the steps involved not only helps in solving the specific equation but also builds a strong foundation for tackling more complex problems in the future. The journey from the initial equation to the final answer showcases the power of algebraic techniques in uncovering the underlying solutions. This step-by-step guide provides a clear and concise method for solving quadratic equations, ensuring clarity and accuracy in the problem-solving process. With the solutions clearly identified and verified, we have successfully completed the task.

Conclusion

In conclusion, solving the quadratic equation $x^2 - 1 = 399$ has been a valuable exercise in understanding algebraic techniques. We successfully identified the solutions as $x = 20$ and $x = -20$ by following a systematic approach: first, we isolated the $x^2$ term, then we took the square root of both sides (remembering to consider both positive and negative roots), and finally, we verified our solutions to ensure accuracy. This process underscores the importance of a methodical approach to problem-solving in mathematics. By breaking down complex problems into smaller, manageable steps, we can confidently arrive at the correct answers. Furthermore, this exercise highlights the significance of understanding the underlying principles of algebra, such as the properties of equations and the implications of taking square roots. Mastering these concepts not only helps in solving equations but also builds a strong foundation for more advanced mathematical studies. The ability to solve quadratic equations is a fundamental skill that has applications in various fields, including physics, engineering, and computer science. Therefore, understanding and practicing these techniques is crucial for anyone pursuing studies or careers in these areas. This step-by-step guide serves as a valuable resource for students and anyone seeking to improve their algebraic problem-solving skills. With practice and a clear understanding of the methods, solving quadratic equations can become a straightforward and rewarding endeavor.