Solving $x^2 = 54$ A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the equation $x^2 = 54$ where $x$ is a real number. Our primary objective is to simplify the solution as much as possible and present it in a clear, concise manner. This involves understanding the properties of square roots and how they interact with algebraic equations. We will explore the concept of finding both positive and negative roots, and the importance of simplifying radicals to their simplest form. This exploration is fundamental in algebra and serves as a building block for more complex mathematical problems. Understanding how to solve such equations is crucial for various applications in mathematics, physics, engineering, and other fields. This article aims to provide a step-by-step guide, ensuring that the reader not only understands the solution but also the underlying principles involved. To find the values of $x$ that satisfy the equation, we will take the square root of both sides, remembering to consider both positive and negative roots. We will then simplify the radical to obtain the most concise form of the solution. This process highlights the inverse relationship between squaring a number and taking its square root, a key concept in algebra. So, let’s embark on this mathematical journey to demystify the solution to this equation.

Understanding Square Roots

Before we jump into solving the equation $x^2 = 54$, it's crucial to have a solid grasp of what square roots are. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9. Mathematically, this can be written as $\sqrt{9} = 3$. However, we must also consider the negative root. Since $(-3) \times (-3) = 9$, -3 is also a square root of 9. Therefore, 9 has two square roots: 3 and -3. This dual nature of square roots is a fundamental concept in algebra. It’s important to remember that every positive number has two square roots: a positive square root (also called the principal square root) and a negative square root. This is because squaring either the positive or negative root will result in the same positive number. This understanding is particularly crucial when solving equations involving squares, as we must account for both possibilities. When we encounter an equation like $x^2 = a$, where $a$ is a positive number, we need to consider both the positive and negative square roots of $a$ as potential solutions. This principle forms the basis for solving many algebraic problems and is a cornerstone of mathematical understanding. Failing to consider both roots can lead to incomplete or incorrect solutions, especially in more complex equations and real-world applications. So, let’s keep this concept in mind as we proceed to solve our equation.

Step-by-Step Solution

To solve the equation $x^2 = 54$, we need to isolate $x$. The most direct way to do this is by taking the square root of both sides of the equation. This process will help us undo the square on the $x$ term and reveal the possible values of $x$. Remember, when taking the square root of both sides, it is crucial to consider both the positive and negative roots, as discussed earlier. This is because both a positive and a negative number, when squared, will result in a positive value. So, we have: $\sqrt{x^2} = \pm \sqrt{54}$. The square root of $x^2$ is simply $x$, so we get $x = \pm \sqrt{54}$. Now, we need to simplify $\sqrt{54}$. To simplify a square root, we look for perfect square factors within the number under the radical. In this case, 54 can be factored as $9 \times 6$, where 9 is a perfect square ($3^2 = 9$). Thus, we can rewrite $\sqrt{54}$ as $\sqrt{9 \times 6}$. Using the property of square roots that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can further simplify this to $\sqrt{9} \times \sqrt{6}$. Since $\sqrt{9} = 3$, we have $3\sqrt{6}$. Therefore, the solutions for $x$ are $x = \pm 3\sqrt{6}$. This means there are two solutions: $x = 3\sqrt{6}$ and $x = -3\sqrt{6}$. By following these steps, we have successfully solved the equation and simplified the answer.

Simplifying the Radical

Simplifying radicals is a crucial step in solving equations like $x^2 = 54$. The goal of simplifying a radical is to express it in its simplest form, where the number under the square root has no perfect square factors other than 1. In our case, we started with $\sqrt{54}$. To simplify this, we need to find the largest perfect square that divides 54. As we saw earlier, 54 can be factored as $9 \times 6$. The number 9 is a perfect square because it is $3^2$. Therefore, we can rewrite $\sqrt{54}$ as $\sqrt{9 \times 6}$. Using the property of square roots, $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate this into $\sqrt{9} \times \sqrt{6}$. We know that $\sqrt{9} = 3$, so we have $3\sqrt{6}$. Now, we need to check if $\sqrt{6}$ can be simplified further. The factors of 6 are 1, 2, 3, and 6. None of these (other than 1) are perfect squares, so $\sqrt{6}$ is in its simplest form. Therefore, the simplified form of $\sqrt{54}$ is $3\sqrt{6}$. This process of breaking down the number under the radical into its factors, identifying perfect squares, and extracting their square roots is essential for simplifying radicals. Simplifying radicals not only makes the answer more concise but also makes it easier to work with in further calculations. This skill is fundamental in algebra and is frequently used in various mathematical contexts. By simplifying radicals, we ensure that our answers are in their most elegant and usable form.

Final Answer

In summary, to solve the equation $x^2 = 54$, we followed a series of steps that involved understanding square roots, applying algebraic principles, and simplifying radicals. We began by taking the square root of both sides of the equation, remembering to consider both positive and negative roots. This gave us $x = \pm \sqrt{54}$. We then focused on simplifying the radical $\sqrt{54}$. By identifying the perfect square factor of 9 within 54, we were able to rewrite $\sqrt{54}$ as $\sqrt{9 \times 6}$. Applying the property of square roots, we separated this into $\sqrt{9} \times \sqrt{6}$, which simplified to $3\sqrt{6}$. Thus, we arrived at the solutions $x = \pm 3\sqrt{6}$. This means there are two solutions: $x = 3\sqrt{6}$ and $x = -3\sqrt{6}$. These are the values of $x$ that, when squared, equal 54. We have simplified the solutions as much as possible, ensuring that the radical is in its simplest form. This process demonstrates the importance of understanding the properties of square roots and the techniques for simplifying radicals. It also highlights the significance of considering both positive and negative roots when solving equations involving squares. Therefore, the final answer to the equation $x^2 = 54$ is $x = 3\sqrt{6}, -3\sqrt{6}$. These solutions represent the complete and simplified answer to the problem, showcasing the application of algebraic principles and radical simplification techniques.