In this article, we'll walk through the process of solving the quadratic equation x² + 4x = 5 by using the method of completing the square. This technique is a powerful tool for solving quadratic equations, especially when they are not easily factorable. We will break down each step with detailed explanations and examples to ensure a clear understanding.
Understanding the Method of Completing the Square
Completing the square is a technique used to convert a quadratic equation in the form ax² + bx + c = 0 into the form (x + p)² = q, where p and q are constants. This form makes it easy to solve for x by taking the square root of both sides. The main idea behind this method is to manipulate the quadratic equation so that one side becomes a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as (x + a)² or (x - a)². Completing the square involves adding a constant term to both sides of the equation to create this perfect square trinomial. This method is particularly useful when the quadratic equation cannot be easily factored using traditional methods. It allows us to rewrite the equation in a form that is readily solvable by isolating the variable x. Understanding the underlying principles of completing the square is crucial for mastering this technique. It not only provides a method for solving quadratic equations but also enhances problem-solving skills in algebra. By following the steps meticulously and practicing with various examples, you can become proficient in using this method to solve a wide range of quadratic equations. The beauty of completing the square lies in its systematic approach, ensuring that you can always find a solution regardless of the complexity of the equation. Each step is designed to bring the equation closer to a solvable form, making it a valuable tool in your mathematical arsenal. This method also lays the groundwork for understanding the quadratic formula, which is derived from the process of completing the square itself. Therefore, mastering this technique is essential for a comprehensive understanding of quadratic equations and their solutions. — Chelsea Vs West Ham: Premier League Showdown
Step-by-Step Solution of x² + 4x = 5
Let's dive into solving the equation x² + 4x = 5. This equation is a quadratic equation, which means it involves a term with x raised to the power of 2. Our goal is to find the values of x that satisfy this equation. The method of completing the square will help us rewrite the equation in a form that makes it easier to solve. First, we need to make sure the equation is in the standard form for completing the square. In this case, the equation is already set up nicely with the quadratic and linear terms on one side and the constant term on the other. The next crucial step is to find the value that we need to add to both sides of the equation to complete the square. This value is determined by taking half of the coefficient of the x term (which is 4 in our case), squaring it, and then adding the result to both sides. Half of 4 is 2, and 2 squared is 4. So, we add 4 to both sides of the equation. This gives us x² + 4x + 4 = 5 + 4. The left side of this equation is now a perfect square trinomial. It can be factored into (x + 2)². The right side simplifies to 9. Thus, our equation becomes (x + 2)² = 9. Now, we can take the square root of both sides to eliminate the square on the left side. Remember that when we take the square root of a number, we need to consider both the positive and negative roots. The square root of 9 is both 3 and -3. So, we have two equations to solve: x + 2 = 3 and x + 2 = -3. Solving x + 2 = 3, we subtract 2 from both sides to get x = 1. Solving x + 2 = -3, we subtract 2 from both sides to get x = -5. Therefore, the solutions to the equation x² + 4x = 5 are x = 1 and x = -5. These are the values of x that, when substituted back into the original equation, will make the equation true. We have successfully used the method of completing the square to find these solutions.
Step 1: Ensure the Coefficient of x² is 1
In our equation, x² + 4x = 5, the coefficient of x² is already 1, which is perfect for proceeding with the method of completing the square. If the coefficient were not 1, we would need to divide the entire equation by that coefficient to make it 1. This step is crucial because the method of completing the square works most efficiently when the leading coefficient is 1. If the coefficient of x² is not 1, dividing the equation by that coefficient ensures that the subsequent steps are performed correctly. For instance, if we had an equation like 2x² + 8x = 10, we would first divide the entire equation by 2, resulting in x² + 4x = 5, which is the equation we are currently working with. This preliminary step simplifies the process and allows us to apply the core principles of completing the square without any additional complications. Ensuring the leading coefficient is 1 sets the stage for accurately calculating the constant term needed to complete the square. It also aligns the equation with the standard form required for applying the method effectively. By skipping this step when necessary, you risk obtaining incorrect solutions. Therefore, always double-check the coefficient of x² and make sure it is 1 before moving forward with the method. This simple check can save you a lot of time and prevent errors in your solution. The importance of this step cannot be overstated, as it forms the foundation for the entire process of completing the square. It is a fundamental aspect that ensures the rest of the steps will lead to the correct solution. So, remember to always prioritize this step and make it a habit to verify the coefficient of x² before proceeding with the rest of the method.
Step 2: Move the Constant Term to the Right Side
In the equation x² + 4x = 5, the constant term is already isolated on the right side of the equation, which simplifies our work. This step is about rearranging the equation so that the terms involving x are on one side and the constant term is on the other. When the constant term is not already on the right side, you would typically add or subtract it from both sides to move it. For example, if we had an equation like x² + 4x - 5 = 0, we would add 5 to both sides to get x² + 4x = 5. This rearrangement prepares the equation for the next crucial step of completing the square, which involves adding a specific value to both sides to create a perfect square trinomial. The goal here is to set up the equation in a format that allows us to easily manipulate it into the form (x + p)² = q. By isolating the constant term, we create a clear distinction between the variable terms and the constant, making the subsequent steps more straightforward. This step is not just about moving numbers around; it’s about organizing the equation in a way that facilitates the completing the square process. A well-organized equation is easier to work with and reduces the chances of making mistakes. Therefore, ensuring the constant term is on the right side is a key step in the overall strategy of solving quadratic equations by completing the square. This preparatory step makes the rest of the process smoother and more manageable, allowing you to focus on the core technique of creating a perfect square trinomial. By ensuring this separation, you are setting the stage for a successful application of the completing the square method.
Step 3: Calculate the Value to Complete the Square
To complete the square for the equation x² + 4x = 5, we need to find a value that, when added to the left side, will create a perfect square trinomial. This value is calculated by taking half of the coefficient of the x term and squaring it. In our equation, the coefficient of the x term is 4. Half of 4 is 2, and 2 squared is 4. Therefore, the value we need to add to both sides of the equation is 4. This process is fundamental to the method of completing the square. The idea is to transform the quadratic expression into a form that can be easily factored as a square of a binomial. The formula we use is (b/2)², where 'b' is the coefficient of the x term. This calculation ensures that the resulting trinomial on the left side can be expressed as (x + p)², where p is half of the coefficient of x. This step is crucial because it sets up the equation for the next step, which involves factoring the perfect square trinomial and simplifying the equation. Understanding how to calculate this value is essential for mastering the technique of completing the square. It is a systematic way of determining the constant term needed to transform a quadratic expression into a perfect square. This methodical approach allows us to solve quadratic equations that may not be easily factorable by other methods. By carefully calculating this value, we ensure that the equation remains balanced and that the resulting trinomial is indeed a perfect square. This precision is key to the success of the completing the square method. Therefore, pay close attention to this step and make sure you understand the calculation thoroughly.
Step 4: Add the Calculated Value to Both Sides
Having calculated the value to complete the square, which is 4 in our case, we now add this value to both sides of the equation x² + 4x = 5. This gives us x² + 4x + 4 = 5 + 4. Adding the same value to both sides ensures that the equation remains balanced. This step is crucial because it transforms the left side of the equation into a perfect square trinomial, which is the key to solving the equation using this method. The perfect square trinomial can then be factored into the square of a binomial. On the right side, we simply add the numbers to simplify the equation. In this case, 5 + 4 equals 9. So, our equation becomes x² + 4x + 4 = 9. This step bridges the gap between the original equation and the form that is easily solvable. Adding the calculated value is not just a mathematical manipulation; it's a strategic move that allows us to rewrite the equation in a way that isolates the variable x. The balance of the equation is maintained by adding the same value to both sides, which is a fundamental principle in algebra. This careful addition sets the stage for factoring the left side and taking the square root of both sides, leading us closer to the solution. Therefore, this step is a pivotal point in the process of completing the square, paving the way for the final steps in solving the quadratic equation. The strategic addition of this value transforms the equation into a more manageable form, making the solutions accessible.
Step 5: Factor the Perfect Square Trinomial
Now, we factor the perfect square trinomial on the left side of the equation x² + 4x + 4 = 9. The trinomial x² + 4x + 4 can be factored into (x + 2)². This is the essence of completing the square: transforming a quadratic expression into the square of a binomial. Factoring the perfect square trinomial simplifies the equation significantly, making it easier to solve for x. The expression (x + 2)² represents (x + 2) * (x + 2), and when expanded, it gives us x² + 4x + 4. This step is a direct application of the factoring techniques for perfect square trinomials. Recognizing this pattern is crucial for efficiently solving quadratic equations using this method. The factored form (x + 2)² allows us to apply the square root property, which is the next step in solving for x. By rewriting the quadratic expression as a squared binomial, we can eliminate the square by taking the square root of both sides of the equation. This simplification is the core advantage of completing the square, as it transforms a complex quadratic equation into a simpler form that can be readily solved. The ability to recognize and factor perfect square trinomials is a fundamental skill in algebra, and this step highlights its importance in the context of solving quadratic equations. Therefore, mastering this factoring technique is essential for effectively using the method of completing the square.
Step 6: Take the Square Root of Both Sides
With the equation in the form (x + 2)² = 9, we now take the square root of both sides. This step is a critical part of the completing the square method, as it allows us to eliminate the square and isolate the term containing x. When taking the square root of both sides, it's crucial to remember to consider both the positive and negative square roots. The square root of 9 is both 3 and -3. Therefore, we have two possible equations: x + 2 = 3 and x + 2 = -3. This recognition of both positive and negative roots is essential for finding all possible solutions to the quadratic equation. Neglecting the negative root would lead to missing one of the solutions. Taking the square root of both sides is a direct application of the properties of square roots and squares. It undoes the squaring operation, bringing us closer to solving for x. This step is a turning point in the solution process, as it transforms the equation from a squared form to a linear form, which is much easier to solve. The introduction of both positive and negative roots reflects the nature of quadratic equations, which can have up to two distinct real solutions. Therefore, being mindful of both roots ensures a complete and accurate solution. This step demonstrates the power of inverse operations in algebra, where we use the square root to reverse the squaring operation. By carefully considering both positive and negative roots, we set the stage for finding all the values of x that satisfy the original equation. — SI Unit For Temperature In Gas Law Calculations Exploring Kelvin
Step 7: Solve for x
Now we have two simple linear equations to solve: x + 2 = 3 and x + 2 = -3. These equations are derived from taking the square root of both sides of the equation in the previous step. To solve for x in each equation, we subtract 2 from both sides. For the first equation, x + 2 = 3, subtracting 2 from both sides gives us x = 3 - 2, which simplifies to x = 1. For the second equation, x + 2 = -3, subtracting 2 from both sides gives us x = -3 - 2, which simplifies to x = -5. Therefore, the solutions to the original quadratic equation x² + 4x = 5 are x = 1 and x = -5. These values of x are the roots of the equation, and they are the points where the parabola represented by the quadratic equation intersects the x-axis. Solving these linear equations is a straightforward application of algebraic principles. The goal is to isolate x on one side of the equation, which we achieve by performing the inverse operation of addition (which is subtraction). These final steps demonstrate the culmination of the completing the square method. By systematically transforming the original quadratic equation, we have arrived at two simple linear equations that can be easily solved. The solutions x = 1 and x = -5 represent the complete solution set for the quadratic equation. This step underscores the power of completing the square as a technique for solving quadratic equations that may not be easily factorable by other methods. The ability to break down a complex problem into simpler steps is a key skill in mathematics, and this step exemplifies that approach. — Dodge Dakota RT For Sale: Your Ultimate Guide
Solution
The solution set for the equation x² + 4x = 5 is {1, -5}. This means that when we substitute either 1 or -5 for x in the original equation, the equation holds true. We have successfully solved the equation by completing the square, a method that involves transforming the quadratic equation into a perfect square trinomial. This method is particularly useful when the quadratic equation is not easily factorable by simple inspection. The solutions we found, 1 and -5, are the roots of the equation, and they represent the x-intercepts of the parabola defined by the quadratic equation. These values are the points where the parabola crosses the x-axis on a graph. Verifying these solutions by substituting them back into the original equation is a good practice to ensure accuracy. When x = 1, the equation becomes 1² + 4(1) = 5, which simplifies to 1 + 4 = 5, which is true. When x = -5, the equation becomes (-5)² + 4(-5) = 5, which simplifies to 25 - 20 = 5, which is also true. Therefore, we have confirmed that our solutions are correct. The process of completing the square not only provides the solutions to the equation but also offers valuable insights into the structure and properties of quadratic equations. Understanding this method enhances problem-solving skills and provides a solid foundation for more advanced algebraic concepts. The solution set {1, -5} represents the complete set of values for x that satisfy the given quadratic equation.
Conclusion
In conclusion, we have successfully solved the quadratic equation x² + 4x = 5 by using the method of completing the square. This technique is a valuable tool in algebra, allowing us to find the solutions to quadratic equations that may not be easily factorable. The step-by-step process we followed ensures a clear understanding of the method and its application. We started by ensuring the coefficient of x² was 1, then moved the constant term to the right side of the equation. Next, we calculated the value needed to complete the square, which was 4 in this case, and added it to both sides of the equation. This transformed the left side into a perfect square trinomial, which we then factored into (x + 2)². Taking the square root of both sides gave us two linear equations, which we solved to find the solutions x = 1 and x = -5. The solution set {1, -5} represents the values of x that satisfy the original equation. Completing the square is not only a method for solving quadratic equations but also a foundational concept in algebra. It provides a systematic approach to transforming quadratic expressions and equations into forms that are easier to work with. Mastering this technique enhances problem-solving skills and provides a deeper understanding of quadratic functions. This method also lays the groundwork for understanding the quadratic formula, which is derived from the process of completing the square itself. Therefore, the ability to complete the square is an essential skill for anyone studying algebra and beyond. The solutions obtained through this method are accurate and reliable, providing a solid foundation for further mathematical explorations.
