Simplify √x²-18x: A Step-by-Step Guide

In the realm of mathematics, simplifying expressions involving radicals is a fundamental skill. When confronted with the expression √x²-18x, it's crucial to understand the steps involved in extracting the square root and presenting the expression in its most simplified form. This process involves identifying perfect square factors within the radicand (the expression under the radical) and extracting their square roots. This comprehensive guide will walk you through the intricacies of simplifying √x²-18x, providing you with a clear understanding of the underlying principles and techniques.

To embark on this mathematical journey, we'll first delve into the core concept of simplifying radicals. A radical expression is considered simplified when the radicand has no perfect square factors other than 1, no fractions under the radical sign, and no radicals in the denominator of a fraction. Achieving this simplified form often involves factoring the radicand and extracting any perfect square factors. For instance, √16 can be simplified to 4 because 16 is a perfect square (4²). Similarly, √75 can be simplified to 5√3 by factoring 75 into 25 x 3, where 25 is a perfect square.

The expression √x²-18x presents a slightly more complex challenge due to the presence of the variable x. However, the underlying principle of identifying perfect square factors remains the same. To simplify this expression, we need to employ a technique called completing the square. This technique involves manipulating the expression under the radical to create a perfect square trinomial, which can then be easily factored and simplified. Completing the square is a powerful tool in algebra and calculus, enabling us to solve quadratic equations, graph parabolas, and, as in this case, simplify radical expressions. In order to proceed, let's go through each of the steps involved in simplifying this expression, providing detailed explanations and examples along the way. This approach will empower you to confidently tackle similar simplification problems in the future.

Step-by-Step Simplification of √x²-18x

The simplification of √x²-18x involves a strategic application of algebraic techniques, primarily completing the square. This method allows us to rewrite the expression under the radical in a form that reveals its perfect square components. Let's break down the process into manageable steps:

Step 1: Factoring out a Common Factor

Our initial expression is √x²-18x. The first step in simplifying this expression is to identify and factor out any common factors from the terms under the radical. In this case, both x² and -18x share a common factor of x. Factoring out x, we get:

√x²-18x = √x(x-18)

This factorization is a crucial step as it sets the stage for completing the square. By isolating x as a factor, we can focus on manipulating the expression within the parentheses to create a perfect square trinomial. This initial step highlights the importance of recognizing common factors in algebraic expressions. Factoring not only simplifies expressions but also reveals underlying structures and relationships between terms. It's a fundamental skill that underpins many algebraic manipulations, making it an essential tool in your mathematical arsenal. The next step is all about converting the factored expression into a form suitable for completing the square, which will ultimately lead us to the simplified radical expression.

Step 2: Completing the Square

The heart of simplifying √x²-18x lies in the technique of completing the square. This method allows us to transform the expression x-18 within the radical into a perfect square trinomial, which can then be easily factored. To complete the square, we need to add and subtract a specific constant term within the parentheses. This constant is determined by taking half of the coefficient of the x term (which is -18), squaring it, and then adding and subtracting it within the parentheses. Half of -18 is -9, and (-9)² is 81. Therefore, we add and subtract 81 inside the parentheses:

√x(x-18) = √x(x-18 + 81 - 81)

Now, we can rearrange the terms within the parentheses to group the perfect square trinomial:

√x(x-18 + 81 - 81) = √x((x-18 + 81) - 81)

The expression (x-18 + 81) is now a perfect square trinomial. It can be factored as (x-9)². So, we have:

√x((x-18 + 81) - 81) = √x((x-9)² - 81)

This step is the cornerstone of the simplification process. Completing the square has allowed us to rewrite the expression under the radical in a form that includes a perfect square, which is a prerequisite for further simplification. Understanding the mechanics of completing the square is crucial for mastering this technique. It involves identifying the coefficient of the x term, halving it, squaring the result, and then adding and subtracting this value. This process transforms a quadratic expression into a perfect square trinomial plus a constant term, facilitating factorization and simplification. In the next step, we will see how this transformation helps us further simplify the radical expression.

Step 3: Rewriting the Expression

Following the completion of the square, our expression now looks like this:

√x((x-9)² - 81)

To proceed with the simplification, we need to rewrite the expression to separate the perfect square term. However, it's important to note that we cannot directly distribute the square root over the subtraction within the parentheses. This is because the square root of a difference is not equal to the difference of the square roots (√a-b ≠ √a - √b). Therefore, we need to express the result in terms of separate radicals, taking into consideration the domain of x.

To handle this situation correctly, we can consider two cases:

• Case 1: x ≥ 18

  If x is greater than or equal to 18, then both x and (x-18) are non-negative. In this case, we can rewrite the expression as:

  √x((x-9)² - 81) = √x * √((x-9)² - 81)

  This form separates the square root of x from the square root of the remaining expression. However, we cannot simplify the second radical further without additional information or constraints on the value of x. It's crucial to recognize that the expression under the second radical, (x-9)² - 81, may not always be a perfect square. Therefore, we leave it in this form for now.

• Case 2: 0 ≤ x < 18

  If x is between 0 and 18, then x is non-negative, but (x-18) is negative. This introduces a complication because the square root of a negative number is not a real number. In this case, the original expression √x²-18x is not defined in the real number system. Therefore, there is no real simplification possible for this range of x values.

This step emphasizes the importance of considering the domain of variables when simplifying radical expressions. The domain of a variable refers to the set of all possible values that the variable can take. In this case, the domain of x is restricted by the square root, which requires the radicand to be non-negative. Understanding the domain is crucial for ensuring that the simplification process is mathematically sound and yields valid results. The next step will summarize the simplified forms and highlight the conditions under which they are valid.

Step 4: Final Simplified Form and Conditions

After completing the steps of factoring, completing the square, and rewriting the expression, we arrive at the final simplified form of √x²-18x. However, it's crucial to remember that the simplification depends on the value of x and its domain. Based on our analysis in the previous steps, we can summarize the simplified form and its conditions as follows:

• If x ≥ 18:

  The simplified form is:

  √x((x-9)² - 81) = √x * √((x-9)² - 81)

  In this case, both x and (x-18) are non-negative, and the expression is defined in the real number system. The simplification involves separating the square root of x from the square root of the remaining expression. However, the second radical, √((x-9)² - 81), cannot be simplified further without additional information or constraints on the value of x.

• If 0 ≤ x < 18:

  The expression √x²-18x is not defined in the real number system.

  In this case, x is non-negative, but (x-18) is negative, making the radicand negative. Since the square root of a negative number is not a real number, there is no real simplification possible for this range of x values.

This final step underscores the importance of considering the conditions under which a simplified expression is valid. Simplification in mathematics is not just about manipulating expressions; it's also about preserving their meaning and domain. The conditions we have identified here ensure that the simplified form is equivalent to the original expression within the real number system. It's a crucial aspect of mathematical rigor and accuracy. By explicitly stating the conditions, we provide a complete and unambiguous answer to the simplification problem.

Conclusion

Simplifying √x²-18x is a comprehensive exercise that demonstrates the power of algebraic techniques, particularly completing the square. By factoring out common factors, completing the square, and carefully considering the domain of the variable x, we have successfully simplified the expression and identified the conditions under which the simplified form is valid. This process highlights the importance of a methodical approach to problem-solving in mathematics, where each step builds upon the previous one to arrive at the desired solution. The key takeaway is that simplification is not just about manipulating symbols; it's about understanding the underlying mathematical principles and ensuring that the simplified expression is equivalent to the original one within its defined domain.

Throughout this guide, we have emphasized the significance of understanding the domain of variables, especially when dealing with radicals. The domain restricts the possible values that a variable can take, ensuring that the expression remains defined within the chosen number system (in this case, the real numbers). Ignoring the domain can lead to incorrect simplifications and erroneous results. Therefore, it's always essential to consider the domain when simplifying expressions, especially those involving radicals, fractions, or logarithms.

Furthermore, this exploration has showcased the versatility of completing the square as a technique for simplifying algebraic expressions. Completing the square is not only useful for simplifying radicals but also for solving quadratic equations, graphing parabolas, and performing other algebraic manipulations. It's a fundamental tool in algebra and calculus, and mastering it can significantly enhance your mathematical problem-solving abilities. In conclusion, simplifying √x²-18x is a valuable exercise that reinforces essential algebraic skills and highlights the importance of mathematical rigor and domain awareness. By following the steps outlined in this guide, you can confidently tackle similar simplification problems and deepen your understanding of algebraic manipulations.