Solving 4x² - 72x = 0 By Completing The Square
This article provides a detailed, step-by-step guide to solving the quadratic equation 0 = 4x² - 72x by completing the square. This method is a powerful technique for finding the roots of quadratic equations and offers a deeper understanding of their structure. We will break down each step with clear explanations to ensure you grasp the underlying concepts and can apply this method to various quadratic equations. The completing the square method is a fundamental concept in algebra, providing a systematic approach to rewrite a quadratic expression into a perfect square trinomial plus a constant. This transformation allows us to easily solve for the variable using the square root property. Understanding how to complete the square is crucial for various applications in mathematics, including solving quadratic equations, graphing parabolas, and deriving the quadratic formula. In this article, we will focus on applying this method to the specific equation 0 = 4x² - 72x, illustrating each step with detailed explanations and justifications. By following this guide, you'll gain a solid understanding of the completing the square method and its application in solving quadratic equations.
Understanding Quadratic Equations and the Completing the Square Method
Before we dive into the solution, let's briefly discuss quadratic equations and the completing the square method. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Completing the square is a technique that transforms a quadratic expression into a perfect square trinomial, which is a trinomial that can be factored as (px + q)² or (px - q)². This method is particularly useful when the quadratic equation is not easily factorable. The core idea behind completing the square is to manipulate the equation algebraically to create a perfect square trinomial on one side, allowing us to solve for x by taking the square root of both sides. This method provides a systematic approach to solving quadratic equations, ensuring that we find all possible solutions. By mastering the completing the square technique, you'll gain a deeper understanding of the structure of quadratic equations and develop a valuable tool for solving them. This method is not only useful for solving equations but also for various other applications in mathematics, such as graphing parabolas and deriving the quadratic formula.
Step-by-Step Solution of 0 = 4x² - 72x
Let's now apply the completing the square method to solve the equation 0 = 4x² - 72x. We will break down each step in detail, ensuring that you understand the reasoning behind each operation. This step-by-step approach will help you grasp the completing the square technique and apply it to other quadratic equations. Each step is crucial in transforming the equation into a form that allows us to easily solve for the variable. By carefully following these steps, you'll develop a solid understanding of the completing the square method and its application in solving quadratic equations. This method provides a systematic approach to finding the roots of a quadratic equation, ensuring that you find all possible solutions. Understanding each step is essential for mastering the technique and applying it to various quadratic equations. We will provide detailed explanations and justifications for each step, ensuring that you grasp the underlying concepts. This comprehensive approach will help you build a strong foundation in algebra and enhance your problem-solving skills. The completing the square method is a fundamental concept in mathematics, and mastering it will benefit you in various areas of study.
Step 1: Factor out the Coefficient of x²
The first step in completing the square is to ensure that the coefficient of the x² term is 1. In our equation, 0 = 4x² - 72x, the coefficient of x² is 4. To make it 1, we factor out 4 from the terms on the right side of the equation. This gives us: 0 = 4(x² - 18x). Factoring out the coefficient of x² simplifies the equation and allows us to proceed with the completing the square process. This step is crucial because it ensures that the quadratic expression inside the parentheses is in the standard form required for completing the square. By factoring out the coefficient, we isolate the x² term and the x term, making it easier to manipulate the equation and create a perfect square trinomial. This step is a fundamental part of the completing the square method, and mastering it will help you solve a wide range of quadratic equations. Understanding the reasoning behind this step is essential for grasping the overall concept of completing the square and applying it effectively. This step sets the stage for the subsequent steps, which involve manipulating the equation to create a perfect square trinomial.
Step 2: Complete the Square
Now, we focus on the expression inside the parentheses: x² - 18x. To complete the square, we need to add and subtract a specific value. This value is determined by taking half of the coefficient of the x term, squaring it, and adding it to the expression. In this case, the coefficient of the x term is -18. Half of -18 is -9, and (-9)² is 81. So, we add and subtract 81 inside the parentheses: 0 = 4(x² - 18x + 81 - 81). Adding and subtracting the same value doesn't change the equation's overall value, but it allows us to create a perfect square trinomial. The perfect square trinomial is formed by the terms x² - 18x + 81, which can be factored as (x - 9)². This step is the heart of the completing the square method, as it transforms the quadratic expression into a form that can be easily solved. By adding and subtracting the appropriate value, we create a perfect square trinomial that allows us to rewrite the equation in a more manageable form. This step requires careful attention to detail, as the value added and subtracted must be calculated correctly to ensure that the equation remains balanced. Mastering this step is crucial for successfully completing the square and solving quadratic equations.
Step 3: Rewrite as a Perfect Square Trinomial
Now, we rewrite the expression inside the parentheses using the perfect square trinomial we created. We have: 0 = 4((x - 9)² - 81). The trinomial x² - 18x + 81 has been replaced with its factored form, (x - 9)². This step simplifies the equation and brings us closer to isolating the variable x. By rewriting the expression as a perfect square trinomial, we transform the quadratic equation into a form that can be easily solved using the square root property. This step is a direct result of the completing the square process, where we manipulated the equation to create a perfect square trinomial. The factored form, (x - 9)², represents the square of a binomial, which simplifies the equation and allows us to proceed with solving for x. This step is crucial for the overall success of the completing the square method, as it sets the stage for the final steps of the solution. Understanding the relationship between the perfect square trinomial and its factored form is essential for mastering this technique. This step highlights the power of completing the square in transforming a quadratic expression into a more manageable form.
Step 4: Distribute and Isolate the Squared Term
Next, we distribute the 4 back into the parentheses: 0 = 4(x - 9)² - 324. Then, we isolate the squared term by adding 324 to both sides of the equation: 324 = 4(x - 9)². This step prepares the equation for taking the square root of both sides. By distributing the 4 and isolating the squared term, we bring the equation closer to the form where we can easily solve for x. This step involves algebraic manipulation to isolate the (x - 9)² term, which is crucial for applying the square root property. Adding 324 to both sides of the equation maintains the equality while moving the constant term to the left side. This step is an important part of the completing the square method, as it isolates the squared term and prepares the equation for the final steps of the solution. Understanding the importance of maintaining equality while manipulating the equation is essential for mastering this technique. This step demonstrates the power of algebraic manipulation in solving quadratic equations.
Step 5: Divide and Take the Square Root
Now, divide both sides of the equation by 4: 81 = (x - 9)². To solve for x, we take the square root of both sides: ±√81 = x - 9. This gives us ±9 = x - 9. Remember that when taking the square root, we must consider both positive and negative solutions. Taking the square root of both sides is a crucial step in solving the equation, as it allows us to eliminate the square and isolate the variable x. This step highlights the importance of considering both positive and negative roots, as quadratic equations typically have two solutions. The square root property states that if a² = b, then a = ±√b. Applying this property correctly is essential for finding all possible solutions to the equation. This step is a direct consequence of the completing the square process, where we transformed the equation into a form that allows us to easily apply the square root property. Understanding the square root property is crucial for mastering this technique.
Step 6: Solve for x
Finally, we solve for x by adding 9 to both sides of the equation: x = 9 ± 9. This gives us two possible solutions: x = 9 + 9 = 18 and x = 9 - 9 = 0. Therefore, the solutions to the equation 0 = 4x² - 72x are x = 0 and x = 18. This is the final step in the completing the square method, where we isolate the variable x and find its possible values. Adding 9 to both sides of the equation isolates x and provides the solutions to the quadratic equation. The two solutions, x = 0 and x = 18, represent the roots or zeros of the quadratic equation. This step demonstrates the effectiveness of the completing the square method in finding the solutions to quadratic equations. By carefully following each step, we have successfully solved the equation and found the values of x that satisfy the equation. This step is the culmination of the completing the square process, where we transformed the equation, applied the square root property, and isolated the variable x. Understanding this step is crucial for mastering the completing the square technique.
Final Answer
The solutions to the equation 0 = 4x² - 72x are x = 0 and x = 18. Therefore, the correct answer is C. x = 0, 18. This article has provided a detailed, step-by-step guide to solving this quadratic equation by completing the square. By understanding each step and the reasoning behind it, you can apply this method to solve a variety of quadratic equations. The completing the square method is a valuable tool in algebra, providing a systematic approach to finding the roots of quadratic equations. Mastering this technique will enhance your problem-solving skills and provide a deeper understanding of quadratic equations. The final answer, x = 0 and x = 18, represents the values of x that satisfy the original equation. These solutions can be verified by substituting them back into the equation and confirming that the equation holds true. This article has emphasized the importance of understanding each step of the completing the square process, as this understanding is crucial for applying the method effectively. By following this guide, you have gained a solid understanding of how to solve quadratic equations by completing the square.
