Solving X² + (1/2)x + (1/16) = 4/9 A Step-by-Step Guide
This article delves into the process of solving the quadratic equation x² + (1/2)x + (1/16) = 4/9. We will explore the techniques used to factor perfect-square trinomials and how to apply this factorization to find the solutions for x. This is a fundamental concept in algebra, with applications ranging from basic mathematical problem-solving to more advanced fields like calculus and physics. Understanding the steps involved in solving quadratic equations is crucial for students and anyone working with mathematical models. Let's embark on this journey to unravel the mysteries of this equation and learn the strategies for tackling similar problems.
Identifying a Perfect-Square Trinomial
Our initial equation is x² + (1/2)x + (1/16) = 4/9. The first step in solving this equation efficiently is to recognize that the left side, x² + (1/2)x + (1/16), is a perfect-square trinomial. A perfect-square trinomial is a trinomial that can be factored into the square of a binomial. To determine if a trinomial fits this pattern, we need to check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms. In our case, x² is clearly a perfect square, being the square of x. Similarly, 1/16 is a perfect square as it is the square of 1/4. Now, we need to verify the middle term. The square root of x² is x, and the square root of 1/16 is 1/4. Twice the product of these square roots is 2 * x * (1/4) = (1/2)x, which matches the middle term of our trinomial. Thus, we can confidently say that x² + (1/2)x + (1/16) is indeed a perfect-square trinomial. Recognizing this pattern simplifies the solving process significantly, as we can now factor the left side into a binomial squared. This initial identification is a crucial step in efficiently tackling this type of quadratic equation. This technique helps simplify complex equations into manageable forms, paving the way for finding solutions more easily.
Factoring the Perfect-Square Trinomial
Having identified x² + (1/2)x + (1/16) as a perfect-square trinomial, we can now proceed to factor it. The general form for factoring a perfect-square trinomial of the form a² + 2ab + b² is (a + b)². In our equation, x² + (1/2)x + (1/16), we can identify a as x and b as 1/4, since (1/4)² = 1/16. The middle term, (1/2)x, corresponds to 2 * x * (1/4), which confirms our identification of a and b. Therefore, the trinomial x² + (1/2)x + (1/16) can be factored into (x + 1/4)². This factorization is a direct application of the perfect-square trinomial identity. The ability to recognize and factor these trinomials is a valuable skill in algebra, allowing for quick and efficient solving of quadratic equations. By rewriting the left side of the equation as (x + 1/4)², we transform the original equation into a simpler form that is easier to manipulate and solve. This step is crucial for isolating x and finding the solutions to the equation. Understanding this factoring process is fundamental for tackling various algebraic problems, making it a key concept for students to grasp. The factored form not only simplifies solving the equation but also provides insights into the structure of the equation and its solutions.
Rewriting the Equation and Preparing to Solve
After factoring the perfect-square trinomial, our original equation x² + (1/2)x + (1/16) = 4/9 transforms into (x + 1/4)² = 4/9. This rewritten equation is significantly simpler to work with than the original. We have successfully condensed the left side into a squared binomial, making the next steps in solving the equation much clearer. The goal now is to isolate x, and the most direct way to do this when dealing with a squared term is to take the square root of both sides of the equation. This process will undo the squaring operation on the left side, bringing us closer to finding the value(s) of x. However, it is crucial to remember that when taking the square root of a number, we must consider both the positive and negative roots. This is because both the positive and negative square roots, when squared, will result in the same positive number. This step is a critical juncture in solving quadratic equations, as it directly leads us to the potential solutions. By preparing the equation in this way, we set the stage for the final steps of finding the values of x that satisfy the original equation. This transformation highlights the power of algebraic manipulation in simplifying complex problems.
Taking the Square Root of Both Sides
To proceed with solving for x in the equation (x + 1/4)² = 4/9, we take the square root of both sides. This operation is a crucial step in isolating x. When we take the square root of (x + 1/4)², we get x + 1/4. On the right side, we need to consider both the positive and negative square roots of 4/9. The square root of 4/9 is ±2/3, since (2/3)² = 4/9 and (-2/3)² = 4/9. Therefore, we have two possible equations: x + 1/4 = 2/3 and x + 1/4 = -2/3. This is a critical point in the solving process, as it acknowledges the two potential solutions that arise from the nature of squaring. The positive and negative roots provide us with two distinct paths to finding the values of x that satisfy the original equation. Recognizing and accounting for both roots is essential for obtaining a complete set of solutions. This step demonstrates the importance of understanding the properties of square roots when solving quadratic equations. By carefully considering both possibilities, we ensure that we do not miss any valid solutions.
Isolating x and Finding the Solutions
Now that we have two equations, x + 1/4 = 2/3 and x + 1/4 = -2/3, we can isolate x in each equation to find the solutions. To isolate x in the first equation, x + 1/4 = 2/3, we subtract 1/4 from both sides. This gives us x = 2/3 - 1/4. To subtract these fractions, we need a common denominator, which is 12. So, we rewrite 2/3 as 8/12 and 1/4 as 3/12. Therefore, x = 8/12 - 3/12 = 5/12. This is our first solution. Now, let's solve the second equation, x + 1/4 = -2/3. Again, we subtract 1/4 from both sides, giving us x = -2/3 - 1/4. Using the same common denominator of 12, we rewrite -2/3 as -8/12 and 1/4 as 3/12. So, x = -8/12 - 3/12 = -11/12. This is our second solution. Thus, the solutions to the equation x² + (1/2)x + (1/16) = 4/9 are x = 5/12 and x = -11/12. These values of x are the points where the quadratic function intersects the line y = 4/9. The process of solving for x involved carefully isolating the variable and performing the necessary arithmetic operations. This step highlights the importance of precision and accuracy in mathematical calculations. By systematically working through each equation, we have successfully found the two solutions to the quadratic equation.
Verifying the Solutions
To ensure the accuracy of our solutions, it is always a good practice to verify them by substituting them back into the original equation. We found the solutions to be x = 5/12 and x = -11/12. Let's first verify x = 5/12. Substituting this value into the original equation, x² + (1/2)x + (1/16) = 4/9, we get (5/12)² + (1/2)(5/12) + 1/16. This simplifies to 25/144 + 5/24 + 1/16. To add these fractions, we need a common denominator, which is 144. Rewriting the fractions, we get 25/144 + 30/144 + 9/144 = 64/144. Simplifying this fraction, we get 4/9, which matches the right side of the original equation. Therefore, x = 5/12 is a valid solution. Now, let's verify x = -11/12. Substituting this value into the original equation, we get (-11/12)² + (1/2)(-11/12) + 1/16. This simplifies to 121/144 - 11/24 + 1/16. Using the common denominator of 144, we rewrite the fractions as 121/144 - 66/144 + 9/144 = 64/144. Simplifying this fraction, we again get 4/9, which confirms that x = -11/12 is also a valid solution. By verifying both solutions, we can be confident in our answers. This step underscores the importance of checking our work to ensure accuracy in solving mathematical equations. Verification not only confirms the correctness of the solutions but also reinforces the understanding of the underlying concepts and processes involved.
Conclusion: Mastering Quadratic Equation Solving
In conclusion, we have successfully navigated the process of solving the quadratic equation x² + (1/2)x + (1/16) = 4/9. We began by recognizing the left side as a perfect-square trinomial, which allowed us to factor it into (x + 1/4)². This simplification was a crucial step in making the equation more manageable. We then took the square root of both sides, remembering to consider both positive and negative roots, which led us to two separate equations. Solving each equation for x, we found the solutions x = 5/12 and x = -11/12. Finally, we verified these solutions by substituting them back into the original equation, confirming their validity. This step-by-step approach highlights the importance of recognizing patterns, applying algebraic principles, and performing accurate calculations. Mastering the techniques for solving quadratic equations is a fundamental skill in mathematics, with applications in various fields. By understanding the concepts and practicing these methods, students and individuals can confidently tackle more complex mathematical problems. This journey through solving this equation serves as a valuable example of how algebraic manipulation and careful attention to detail can lead to accurate and meaningful solutions. The ability to solve quadratic equations is not just about finding the answers; it's about developing a deeper understanding of mathematical relationships and problem-solving strategies.
