Solving X² - 3x = 0 A Comprehensive Guide
In this article, we will delve into the process of solving the quadratic equation x² - 3x = 0. This is a fundamental concept in algebra, and understanding how to solve such equations is crucial for various mathematical and real-world applications. We'll break down the steps involved, ensuring clarity and comprehension for anyone looking to master this skill. Quadratic equations are polynomial equations of the second degree, meaning they contain a term with the variable raised to the power of two. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Solving a quadratic equation involves finding the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, the equation x² - 3x = 0 is particularly amenable to factoring, which is the method we will employ here. Factoring is the process of expressing a quadratic expression as the product of two linear expressions. This method is often the quickest and easiest way to solve quadratic equations, especially when the equation can be factored easily. By mastering this technique, you can efficiently find the solutions to a wide range of quadratic equations. This skill is not only essential for academic success in mathematics but also valuable in various practical situations where mathematical modeling is required. Whether you're a student learning algebra or a professional applying mathematical principles, understanding how to solve quadratic equations is a fundamental and indispensable skill. The ability to manipulate and solve equations is a cornerstone of mathematical literacy, enabling you to tackle more complex problems and make informed decisions based on quantitative analysis. So, let's dive into the specifics of solving x² - 3x = 0 using factoring, and you'll soon see how straightforward and effective this method can be. Remember, practice is key to mastering any mathematical skill, so work through this example carefully, and then try your hand at similar problems to solidify your understanding. With a solid grasp of factoring, you'll be well-equipped to tackle a variety of quadratic equations and confidently apply your knowledge in diverse contexts.
Factoring the Equation
The equation x² - 3x = 0 is a quadratic equation where we aim to find the values of x that make the equation true. To solve this, we'll use the factoring method. Factoring involves rewriting the equation as a product of simpler expressions. In this specific case, we observe that both terms in the equation, x² and -3x, have a common factor, which is 'x'. Factoring out the common factor 'x' allows us to simplify the equation into a more manageable form. This step is crucial because it transforms the quadratic equation into a product of two factors, making it easier to identify the solutions. The process of factoring is based on the distributive property of multiplication over addition and subtraction. When we factor 'x' out of the expression x² - 3x, we are essentially reversing the distributive property. This allows us to rewrite the expression as a product of 'x' and the remaining terms. The result of factoring is the expression x(x - 3) = 0. This form of the equation is significantly easier to analyze and solve. Once we have factored the equation, we can apply the zero-product property, which is a fundamental principle in algebra. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the key to finding the solutions to the equation. By setting each factor equal to zero, we can determine the values of x that satisfy the original equation. The ability to factor quadratic equations is a valuable skill in algebra. It simplifies the process of solving equations and provides a clear understanding of the roots or solutions. Understanding the underlying principles of factoring, such as the distributive property and the zero-product property, is essential for mastering this technique. As you work through various quadratic equations, you will develop a strong intuition for factoring, which will make solving these equations much more efficient. The equation x² - 3x = 0 is a perfect example to illustrate the power and simplicity of factoring. By recognizing the common factor and applying the zero-product property, we can quickly arrive at the solutions. This method is not only efficient but also provides a clear and logical approach to solving quadratic equations. So, let's proceed to set each factor equal to zero and determine the solutions.
Applying the Zero-Product Property
Now that we have factored the equation as x(x - 3) = 0, we can apply the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, the two factors are 'x' and '(x - 3)'. Therefore, for the equation x(x - 3) = 0 to be true, either x must be equal to zero, or (x - 3) must be equal to zero, or both. This principle allows us to break down the original equation into two simpler equations, which are much easier to solve. The zero-product property is a cornerstone of solving equations by factoring. It provides a direct link between the factored form of an equation and its solutions. Understanding and applying this property correctly is crucial for finding the roots of quadratic equations and other polynomial equations. By setting each factor equal to zero, we create two independent equations that can be solved separately. This approach simplifies the problem significantly, allowing us to find the values of x that satisfy the original equation. The first equation we obtain is x = 0. This is a straightforward solution, indicating that one of the roots of the quadratic equation is zero. This means that if we substitute x = 0 back into the original equation, it will hold true. The second equation we obtain is (x - 3) = 0. This equation is also relatively simple to solve. To isolate x, we need to add 3 to both sides of the equation. This gives us x = 3, which is the second root of the quadratic equation. Therefore, we have found two solutions for the equation x² - 3x = 0: x = 0 and x = 3. These are the values of x that make the equation true. To verify these solutions, we can substitute them back into the original equation and check if they satisfy the equation. This step is always a good practice to ensure the accuracy of our solutions. The zero-product property is not only applicable to quadratic equations but also to polynomial equations of higher degrees. Whenever we can factor a polynomial equation into a product of factors, we can use the zero-product property to find its roots. This makes it a powerful tool in algebra and beyond. In summary, applying the zero-product property is a critical step in solving quadratic equations by factoring. It allows us to transform a single equation into multiple simpler equations, making the process of finding the solutions much more manageable. With a solid understanding of this property, you can confidently tackle a wide range of equations and find their roots effectively.
Finding the Solutions
From the application of the zero-product property, we derived two simple equations: x = 0 and x - 3 = 0. The first equation, x = 0, directly gives us one of the solutions. It tells us that when x is equal to 0, the original equation x² - 3x = 0 is satisfied. This is a clear and straightforward solution, and it's important to recognize that zero can indeed be a valid solution for many equations. The second equation, x - 3 = 0, requires a small additional step to isolate x. To do this, we add 3 to both sides of the equation. This maintains the equality and allows us to isolate x on one side. Adding 3 to both sides of x - 3 = 0 gives us x = 3. This is the second solution to the quadratic equation. It indicates that when x is equal to 3, the original equation is also satisfied. Now that we have found both solutions, x = 0 and x = 3, it's a good practice to verify them. This involves substituting each solution back into the original equation x² - 3x = 0 to ensure that they make the equation true. Let's first verify x = 0. Substituting x = 0 into the equation gives us (0)² - 3(0) = 0 - 0 = 0, which is true. This confirms that x = 0 is indeed a solution. Next, let's verify x = 3. Substituting x = 3 into the equation gives us (3)² - 3(3) = 9 - 9 = 0, which is also true. This confirms that x = 3 is also a solution. Therefore, we have successfully found both solutions to the quadratic equation x² - 3x = 0, and we have verified their correctness. The solutions are x = 0 and x = 3. These are the values of x that satisfy the equation, and they represent the roots of the quadratic equation. Finding the solutions to a quadratic equation is a fundamental skill in algebra. It allows us to understand the behavior of the equation and its relationship to the coordinate plane. The solutions represent the x-intercepts of the parabola defined by the quadratic equation. In this case, the parabola intersects the x-axis at the points (0, 0) and (3, 0). Understanding the graphical representation of quadratic equations can provide valuable insights into their solutions. In summary, we have successfully solved the quadratic equation x² - 3x = 0 by factoring and applying the zero-product property. We found two solutions, x = 0 and x = 3, and we verified their correctness by substituting them back into the original equation. This process demonstrates the power and effectiveness of factoring as a method for solving quadratic equations.
Expressing the Solution
Having determined the solutions to the quadratic equation x² - 3x = 0 as x = 0 and x = 3, the final step is to express the solution in the requested format. The instructions specify that we should use a comma to separate answers if there are multiple solutions. In this case, we have two distinct solutions, 0 and 3. Therefore, we will write the solution set as a comma-separated list. This format is a standard way of representing multiple solutions in mathematics. It clearly indicates that both 0 and 3 are values of x that satisfy the original equation. The use of a comma as a separator ensures that the solutions are easily distinguishable and that the entire solution set is presented in a concise and organized manner. When expressing solutions, it's important to adhere to any specific instructions or formatting guidelines provided. This ensures that the answer is presented in a clear and understandable way, and it also demonstrates attention to detail. In mathematics, precision and clarity are crucial, and following formatting conventions is an integral part of effective communication. The solution set {0, 3} represents the complete set of values of x that make the equation x² - 3x = 0 true. There are no other values of x that satisfy this equation. Therefore, this solution set provides a comprehensive answer to the problem. When solving equations, it's always a good idea to double-check that all solutions have been found and that the solution set is complete. This helps to avoid any errors and ensures that the answer is accurate. In this case, we have used factoring and the zero-product property to systematically find all possible solutions to the quadratic equation. The solutions x = 0 and x = 3 are the only values that make the equation true, and we have verified this by substituting them back into the original equation. Therefore, we can confidently express the solution as 0, 3. This is the final answer, presented in the requested format. It represents the culmination of the problem-solving process, from factoring the equation to applying the zero-product property and finally expressing the solutions in a clear and concise manner. In conclusion, the solution to the equation x² - 3x = 0 is 0, 3. This answer is presented in the correct format, with the solutions separated by a comma. The process of solving this equation has demonstrated the power and effectiveness of factoring as a method for finding the roots of quadratic equations. By understanding and applying these techniques, you can confidently tackle a wide range of algebraic problems.
The solution is 0, 3
