Solving X(x-2)≥3 A Step-by-Step Guide To Quadratic Inequalities

In the realm of mathematics, particularly in algebra, quadratic inequalities often present a fascinating challenge. These inequalities, characterized by a variable raised to the power of two, require a systematic approach to solve effectively. This article delves into the intricacies of solving the quadratic inequality x(x-2)≥3, providing a comprehensive, step-by-step guide suitable for students, educators, and anyone with an interest in mathematical problem-solving. Understanding the methods to tackle such problems not only enhances algebraic skills but also lays a strong foundation for more advanced mathematical concepts. Let's embark on this journey of mathematical exploration and unravel the solutions to this intriguing inequality.

Understanding Quadratic Inequalities

To effectively solve the given inequality, it's crucial to first understand what quadratic inequalities are and how they differ from linear inequalities or quadratic equations. A quadratic inequality is a mathematical statement that compares a quadratic expression to another value using inequality symbols such as >, <, ≥, or ≤. Unlike a quadratic equation, which seeks specific values of x that make the expression equal to zero, a quadratic inequality aims to find a range of x values that satisfy the inequality condition. This range often represents an interval or a union of intervals on the number line.

The general form of a quadratic inequality is ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0, where a, b, and c are constants and a ≠ 0. The solutions to these inequalities are the values of x that make the inequality true. The process of finding these solutions involves several key steps, including rearranging the inequality, finding the roots of the corresponding quadratic equation, and testing intervals to determine where the inequality holds true. Understanding the behavior of quadratic functions, especially their parabolic nature, is instrumental in visualizing and interpreting the solutions of quadratic inequalities. The graph of a quadratic function can provide valuable insights into the intervals where the function's values are positive, negative, or zero, which directly corresponds to the solutions of the inequality.

Step 1: Rearrange the Inequality

The initial step in solving the inequality x(x-2)≥3 involves rearranging it into the standard quadratic form. This form, expressed as ax² + bx + c ≥ 0 (or with other inequality signs), allows us to easily identify the coefficients and constant term, which are essential for further steps. To achieve this, we first need to expand the left side of the inequality and then move all terms to one side, leaving zero on the other side. This process ensures that we have a clear quadratic expression to work with and sets the stage for identifying critical points and intervals.

Starting with x(x-2)≥3, we distribute x across the parentheses, resulting in x² - 2x ≥ 3. The next step is to subtract 3 from both sides of the inequality to obtain x² - 2x - 3 ≥ 0. This rearrangement transforms the inequality into the standard quadratic form, where a = 1, b = -2, and c = -3. Having the inequality in this form is crucial because it allows us to apply standard methods for solving quadratic inequalities, such as finding the roots of the corresponding quadratic equation and testing intervals. This step is not just about algebraic manipulation; it's about setting up the problem in a way that makes it solvable using established techniques. The accuracy of this rearrangement is paramount, as any error here will propagate through the rest of the solution process. Therefore, careful attention to detail is essential to ensure the correct quadratic form is obtained.

Step 2: Find the Roots of the Quadratic Equation

After rearranging the inequality into the standard quadratic form, the next critical step is to find the roots of the corresponding quadratic equation. These roots are the values of x that make the quadratic expression equal to zero, and they serve as the boundary points for the intervals where the inequality may hold true. There are several methods to find these roots, including factoring, using the quadratic formula, or completing the square. The choice of method often depends on the specific quadratic expression and personal preference.

For the quadratic expression x² - 2x - 3, we aim to find the values of x that satisfy the equation x² - 2x - 3 = 0. In this case, factoring is a straightforward approach. We look for two numbers that multiply to -3 and add to -2. These numbers are -3 and 1. Thus, we can factor the quadratic expression as (x - 3)(x + 1) = 0. Setting each factor equal to zero gives us the roots x = 3 and x = -1. These roots are crucial because they divide the number line into three intervals: x < -1, -1 < x < 3, and x > 3. The inequality x² - 2x - 3 ≥ 0 may hold true in some or all of these intervals. Alternatively, if factoring were not immediately apparent, the quadratic formula, x = [-b ± √(b² - 4ac)] / (2a), could be used. Substituting a = 1, b = -2, and c = -3 into the formula would also yield the roots x = 3 and x = -1. The accuracy in finding these roots is paramount, as they form the foundation for determining the solution intervals of the inequality.

Step 3: Determine the Intervals and Test Values

With the roots of the quadratic equation identified, the next crucial step involves determining the intervals they define on the number line and testing values within these intervals to ascertain where the inequality holds true. The roots act as critical points, dividing the number line into segments where the quadratic expression's sign remains consistent. By selecting a test value from each interval and substituting it into the original inequality, we can efficiently determine whether the interval satisfies the inequality condition. This process is fundamental to solving quadratic inequalities, as it provides a clear and systematic way to identify the solution set.

For the inequality x² - 2x - 3 ≥ 0, with roots x = -1 and x = 3, the number line is divided into three intervals: (-∞, -1), (-1, 3), and (3, ∞). To determine the validity of the inequality in each interval, we select a test value. For the interval (-∞, -1), we can choose x = -2. Substituting this into the inequality, we get (-2)² - 2(-2) - 3 = 4 + 4 - 3 = 5, which is greater than or equal to 0. Thus, the inequality holds true in this interval. For the interval (-1, 3), we can choose x = 0. Substituting this into the inequality, we get (0)² - 2(0) - 3 = -3, which is less than 0. Thus, the inequality does not hold true in this interval. For the interval (3, ∞), we can choose x = 4. Substituting this into the inequality, we get (4)² - 2(4) - 3 = 16 - 8 - 3 = 5, which is greater than or equal to 0. Thus, the inequality holds true in this interval. This method of testing values is not just a computational exercise; it’s a powerful analytical tool that allows us to understand the behavior of the quadratic expression across different ranges of x values.

Step 4: Write the Solution

Having tested the intervals and determined where the inequality holds true, the final step is to express the solution set in a clear and concise manner. This typically involves using interval notation or set notation to accurately represent the range(s) of x values that satisfy the inequality. The solution should account for whether the endpoints (the roots) are included or excluded, depending on the inequality symbol used (≥, ≤ include the endpoints; >, < exclude them). Presenting the solution in the correct notation is crucial for clear communication of the mathematical result.

Based on our interval testing for x² - 2x - 3 ≥ 0, we found that the inequality holds true for the intervals (-∞, -1) and (3, ∞). Since the inequality symbol is ≥, which includes equality, the roots x = -1 and x = 3 are also part of the solution. Therefore, we include these endpoints in our interval notation. The solution can be written as (-∞, -1] ∪ [3, ∞). This notation indicates that the solution set includes all real numbers less than or equal to -1, as well as all real numbers greater than or equal to 3. The symbol ∪ represents the union of these two intervals, meaning that any x value within either of these ranges will satisfy the original inequality. Alternatively, the solution can be expressed in set notation as {x | x ≤ -1 or x ≥ 3}. This notation reads as “the set of all x such that x is less than or equal to -1 or x is greater than or equal to 3.” Both interval notation and set notation are valid ways to express the solution, and the choice between them often depends on the context or personal preference. The clarity and accuracy of this final step are essential, as it provides the definitive answer to the problem.

Conclusion

Solving quadratic inequalities, such as x(x-2)≥3, is a fundamental skill in algebra that requires a systematic approach. This step-by-step guide has outlined the key processes involved, from rearranging the inequality into standard form and finding the roots of the corresponding quadratic equation, to testing intervals and expressing the solution set. Each step is crucial and builds upon the previous one, ensuring a thorough and accurate solution. Understanding these methods not only enhances problem-solving abilities but also provides a solid foundation for more advanced mathematical concepts. By mastering the techniques presented, students and enthusiasts can confidently tackle a wide range of quadratic inequalities, fostering a deeper appreciation for the power and elegance of mathematical reasoning. The journey through solving this particular inequality serves as a microcosm of the broader mathematical landscape, where careful analysis, logical deduction, and precise execution converge to reveal the underlying truths. As we conclude this exploration, it’s clear that the ability to solve quadratic inequalities is more than just a mathematical skill; it’s a testament to the human capacity for problem-solving and abstract thinking.