Solving X² - 8x + 5 = 0 By Completing The Square Method
Solving quadratic equations is a fundamental skill in algebra, and one powerful technique for doing so is completing the square. This method not only helps find the roots of a quadratic equation but also provides insights into its structure and properties. In this comprehensive guide, we will walk through the process of solving the equation x² - 8x + 5 = 0 using the completing-the-square method. We will break down each step, explain the underlying principles, and offer tips to ensure you understand and master this technique. Whether you're a student tackling algebra problems or someone looking to refresh their math skills, this article will provide a clear and detailed explanation.
Understanding the Completing the Square Method
The method of completing the square is a versatile technique used to solve quadratic equations, transform quadratic expressions, and even derive the quadratic formula. Before diving into our specific equation, let's understand the core idea behind this method. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants. The goal of completing the square is to rewrite the quadratic equation in the form (x - h)² + k = 0, where h and k are also constants. This form is particularly useful because it allows us to easily isolate x and find the solutions. The name "completing the square" comes from the fact that we are manipulating the equation to create a perfect square trinomial on one side. 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. By rewriting our original quadratic equation in this form, we can take the square root of both sides and solve for x. The process involves several key steps: ensuring the coefficient of x² is 1, moving the constant term to the other side of the equation, adding a value to both sides to complete the square, and then solving for x. Each of these steps is crucial, and we will explore them in detail as we solve x² - 8x + 5 = 0. Understanding the principles behind completing the square not only helps in solving quadratic equations but also provides a deeper understanding of algebraic manipulations and problem-solving strategies. This method is not just a mechanical process; it's a powerful tool for understanding the structure of quadratic expressions and their solutions. By mastering this technique, you'll be well-equipped to tackle a variety of algebraic problems and gain a stronger foundation in mathematics.
Step-by-Step Solution for x² - 8x + 5 = 0
Now, let's apply the completing the square method to solve the equation x² - 8x + 5 = 0 step by step. This will provide a clear and detailed illustration of how the method works in practice.
Step 1: Ensure the Coefficient of x² is 1
The first step in completing the square is to make sure that the coefficient of the x² term is 1. In our equation, x² - 8x + 5 = 0, the coefficient of x² is already 1, so we can proceed to the next step. If the coefficient were not 1, we would need to divide the entire equation by that coefficient to make it 1. This is a crucial step because the subsequent steps of the method rely on this condition being met. For example, if we had the equation 2x² - 16x + 10 = 0, we would first divide the entire equation by 2 to get x² - 8x + 5 = 0, which is the equation we are working with. Ensuring the coefficient of x² is 1 simplifies the process of creating a perfect square trinomial. It also aligns with the standard form required for completing the square, making the rest of the steps more straightforward and less prone to errors. This initial check is a fundamental part of the method and should always be the first thing you do when approaching a quadratic equation using completing the square.
Step 2: Move the Constant Term to the Other Side
The next step is to move the constant term to the right side of the equation. In our equation, x² - 8x + 5 = 0, the constant term is 5. To move it to the other side, we subtract 5 from both sides of the equation. This gives us: x² - 8x = -5. Moving the constant term isolates the terms with x on one side, which is necessary for completing the square. This step sets up the equation so that we can focus on creating a perfect square trinomial on the left side. By moving the constant to the right side, we create space on the left side to add the value that will complete the square. This rearrangement is a critical part of the process, as it allows us to manipulate the equation into a form that is more easily solved. The resulting equation, x² - 8x = -5, is now ready for the next step, where we will determine the value needed to complete the square. This step is simple but essential for the overall success of the method.
Step 3: Complete the Square
This is the core step of the method. To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x + a)² or (x - a)². To find the value we need to add, we take half of the coefficient of the x term (which is -8 in our case), square it, and add the result to both sides. Half of -8 is -4, and (-4)² is 16. So, we add 16 to both sides of the equation x² - 8x = -5: x² - 8x + 16 = -5 + 16. Adding the same value to both sides maintains the balance of the equation while transforming the left side into a perfect square trinomial. The left side, x² - 8x + 16, can now be factored as (x - 4)². This is the essence of completing the square – creating a perfect square trinomial that can be easily expressed as the square of a binomial. The right side of the equation simplifies to 11, so our equation becomes (x - 4)² = 11. This step is often the most challenging part of the method, but it is also the most crucial. By correctly completing the square, we transform the quadratic equation into a form that is much easier to solve.
Step 4: Factor the Perfect Square Trinomial
After completing the square, the left side of our equation, x² - 8x + 16, is now a perfect square trinomial. This means it can be factored into the square of a binomial. In our case, x² - 8x + 16 factors to (x - 4)². This factorization is a direct result of the value we added to complete the square. The number we used, 16, was specifically chosen so that the trinomial would factor in this way. The ability to factor the perfect square trinomial is what makes completing the square such a powerful method. It transforms the quadratic expression into a simpler form that is easier to manipulate and solve. Our equation now looks like this: (x - 4)² = 11. The left side is a perfect square, and the right side is a constant. This form of the equation is ideal for solving for x, as we can now take the square root of both sides.
Step 5: Take the Square Root of Both Sides
Now that we have the equation in the form (x - 4)² = 11, we can take the square root of both sides to isolate x. When taking the square root, we must remember to consider both the positive and negative roots. This gives us: √(x - 4)² = ±√11. The square root of (x - 4)² is simply x - 4, and we have x - 4 = ±√11. Considering both the positive and negative square roots is crucial because quadratic equations can have two solutions. The positive and negative roots represent the two possible values of x that satisfy the original equation. This step is a key transition in the process, as it moves us from a squared term to a linear term, bringing us closer to solving for x. The equation x - 4 = ±√11 now only requires one more step to isolate x completely.
Step 6: Solve for x
The final step is to isolate x by adding 4 to both sides of the equation x - 4 = ±√11. This gives us: x = 4 ± √11. This means we have two solutions for x: x = 4 + √11 and x = 4 - √11. These are the roots of the quadratic equation x² - 8x + 5 = 0. The two solutions represent the points where the parabola described by the quadratic equation intersects the x-axis. These solutions are exact values, expressed in terms of the square root of 11. If we need decimal approximations, we can use a calculator to find that √11 is approximately 3.3166. Therefore, the approximate solutions are x ≈ 4 + 3.3166 ≈ 7.3166 and x ≈ 4 - 3.3166 ≈ 0.6834. The solutions we have found are the values of x that make the equation x² - 8x + 5 = 0 true. By completing the square, we have successfully solved the quadratic equation and found its roots.
Conclusion
In this article, we have walked through the completing the square method to solve the quadratic equation x² - 8x + 5 = 0. Each step was carefully explained, from ensuring the coefficient of x² is 1 to isolating x and finding the two solutions. Completing the square is a powerful technique that not only solves quadratic equations but also provides a deeper understanding of their structure. By mastering this method, you gain a valuable tool for tackling algebraic problems and strengthening your mathematical skills. Whether you are a student learning algebra or someone refreshing your math knowledge, the ability to complete the square is a valuable asset. It provides a method for solving quadratic equations that is both reliable and insightful, offering a clear pathway to the solutions. We encourage you to practice this method with other quadratic equations to further solidify your understanding and skills. The more you practice, the more comfortable and confident you will become in applying this technique to a variety of problems. The process of completing the square is not just about finding solutions; it is also about understanding the mathematical principles behind those solutions. By understanding the method, you gain a deeper appreciation for the elegance and power of algebra.
