Solving X² - 16x + 64 = 24 A Step-by-Step Guide
In this comprehensive guide, we'll walk through the process of solving the quadratic equation x² - 16x + 64 = 24. Quadratic equations are a fundamental concept in algebra, and mastering their solution is crucial for various mathematical and real-world applications. This article aims to provide a clear, step-by-step approach to solving this specific equation, ensuring that you understand each stage of the process. We'll cover essential techniques like simplifying the equation, factoring, and using the square root property. By the end of this guide, you'll be well-equipped to tackle similar quadratic equations with confidence. This method not only helps in solving mathematical problems but also enhances logical reasoning and problem-solving skills, which are valuable in various aspects of life. Let's dive in and explore the solution together. By understanding each step thoroughly, you'll not only solve this particular equation but also gain a solid foundation for tackling more complex problems in the future. Remember, practice is key to mastering these concepts, so work through each step carefully and try applying the same methods to other quadratic equations. This will solidify your understanding and build your confidence in solving such problems.
Understanding Quadratic Equations
Before we dive into the solution, let's briefly discuss what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These equations appear in numerous contexts, from physics and engineering to economics and computer science. Understanding how to solve them is a fundamental skill in mathematics. The solutions to a quadratic equation are also known as its roots or zeros, and they represent the x-values where the parabola described by the equation intersects the x-axis. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages, and the choice of method often depends on the specific form of the equation. In this guide, we'll primarily focus on factoring and using the square root property, as they are particularly well-suited to the given equation. However, it's beneficial to be familiar with all the methods to have a versatile toolkit for solving various types of quadratic equations. Each technique provides a different perspective and can be more efficient depending on the characteristics of the equation.
Step 1: Simplify the Equation
The first step in solving x² - 16x + 64 = 24 is to simplify the equation by setting it equal to zero. This is a crucial step because most methods for solving quadratic equations, such as factoring and using the quadratic formula, require the equation to be in the standard form ax² + bx + c = 0. To do this, we subtract 24 from both sides of the equation:
x² - 16x + 64 - 24 = 24 - 24 x² - 16x + 40 = 0
Now, the equation is in the standard quadratic form. Simplifying the equation in this way makes it easier to identify the coefficients a, b, and c, which are essential for applying various solving methods. In this case, a = 1, b = -16, and c = 40. Having the equation in standard form also allows us to recognize patterns and determine the most efficient method for finding the roots. For instance, if the equation can be easily factored, this method might be preferred. Alternatively, if the coefficients are more complex, the quadratic formula might be a better choice. Understanding this preliminary step is crucial for successfully navigating the rest of the solution process. It lays the foundation for applying the appropriate techniques and accurately determining the roots of the equation. Furthermore, it ensures that we are working with a form of the equation that is conducive to further manipulation and analysis.
Step 2: Attempt to Factor the Quadratic
Factoring is often the quickest way to solve a quadratic equation, provided the equation can be factored easily. We look for two numbers that multiply to give the constant term (40) and add up to the coefficient of the x term (-16). Let's consider the factors of 40:
- 1 and 40
- 2 and 20
- 4 and 10
- 5 and 8
None of these pairs add up to -16. This indicates that the quadratic expression x² - 16x + 40 cannot be factored using simple integers. When simple factoring doesn't work, it doesn't mean the equation has no solution; it simply means we need to explore other methods. Recognizing when an equation is not easily factorable is a crucial skill in solving quadratic equations. It saves time and guides us toward more appropriate solution techniques. In this case, since factoring doesn't seem straightforward, we will move on to another method. However, before we proceed, it's worth noting that some equations may require more advanced factoring techniques, such as factoring by grouping or using complex numbers. But for the scope of this guide, we will focus on methods that are more generally applicable when simple factoring fails.
Step 3: Use the Quadratic Formula
Since we couldn't factor the equation, we'll use the quadratic formula, which is a reliable method for finding the roots of any quadratic equation. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
For our equation x² - 16x + 40 = 0, we have a = 1, b = -16, and c = 40. Plugging these values into the formula gives us:
x = (-(-16) ± √((-16)² - 4 * 1 * 40)) / (2 * 1) x = (16 ± √(256 - 160)) / 2 x = (16 ± √96) / 2
The quadratic formula is a powerful tool because it provides a direct method for finding the solutions, regardless of whether the equation is factorable. It's derived from the process of completing the square and is a fundamental concept in algebra. Understanding the quadratic formula not only allows you to solve a wide range of equations but also deepens your understanding of the relationship between the coefficients and the roots of a quadratic equation. Moreover, the discriminant (b² - 4ac) within the formula provides valuable information about the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. In our case, the discriminant is 96, which is positive, indicating that we will have two distinct real roots.
Step 4: Simplify the Square Root
Now, let's simplify the square root. We can simplify √96 by finding the prime factorization of 96:
96 = 2 * 48 = 2 * 2 * 24 = 2 * 2 * 2 * 12 = 2 * 2 * 2 * 2 * 6 = 2 * 2 * 2 * 2 * 2 * 3
So, √96 = √(2⁵ * 3) = √(2⁴ * 2 * 3) = 2²√6 = 4√6
Substituting this back into our equation, we get:
x = (16 ± 4√6) / 2
Simplifying square roots is a crucial skill in algebra, as it allows us to express solutions in their simplest form. This not only makes the solutions easier to understand and work with but also reveals the underlying mathematical structure more clearly. The process of simplifying square roots often involves identifying perfect square factors within the radicand (the number under the square root) and extracting them. In this case, we found that 96 has a perfect square factor of 16 (2⁴), which allowed us to simplify √96 to 4√6. This simplification step is not just about making the expression look neater; it's about expressing the solution in its most precise and understandable form. It also highlights the importance of understanding number theory concepts such as prime factorization and perfect squares in algebraic manipulations. By simplifying radicals, we ensure that our solutions are accurate, concise, and easily comparable to other mathematical expressions.
Step 5: Final Simplification
Finally, we can simplify the expression further by dividing both terms in the numerator by 2:
x = (16 ± 4√6) / 2 x = 8 ± 2√6
Thus, the two solutions for x are:
x₁ = 8 + 2√6 x₂ = 8 - 2√6
These are the roots of the quadratic equation x² - 16x + 64 = 24. Final simplification is an essential step in solving mathematical problems. It ensures that the answer is presented in its most concise and understandable form. In this case, by dividing both terms in the numerator by the common factor of 2, we simplified the expression from (16 ± 4√6) / 2 to 8 ± 2√6. This not only makes the solutions look cleaner but also makes them easier to interpret and use in further calculations. Simplification is not just a cosmetic step; it's a crucial part of the problem-solving process. It demonstrates a thorough understanding of mathematical principles and attention to detail. Moreover, simplified solutions are less prone to errors when used in subsequent steps of a problem or in related applications. The ability to simplify expressions effectively is a valuable skill in mathematics, highlighting a comprehensive grasp of algebraic manipulations and numerical relationships.
Summary
In summary, we solved the quadratic equation x² - 16x + 64 = 24 by first simplifying the equation to standard form, attempting to factor (which didn't work in this case), applying the quadratic formula, simplifying the square root, and then performing final simplifications. The two solutions we found are x₁ = 8 + 2√6 and x₂ = 8 - 2√6. This step-by-step process demonstrates a systematic approach to solving quadratic equations, which can be applied to a wide range of similar problems. Each step, from simplifying the equation to final simplification, is crucial in ensuring an accurate and understandable solution. Understanding these steps not only helps in solving mathematical problems but also enhances problem-solving skills in various fields. The ability to break down a complex problem into smaller, manageable steps is a valuable asset, whether in mathematics or in real-world situations. Moreover, this process reinforces key mathematical concepts, such as the quadratic formula, simplifying radicals, and the importance of precision in calculations. By mastering these techniques, you gain confidence in your mathematical abilities and are better equipped to tackle more challenging problems in the future. Remember, practice is key to solidifying your understanding and developing proficiency in solving quadratic equations.
