Solving Z² - 4z + 4 = 0 Find The Values Of Z

In this article, we will delve into the process of finding the possible value or values of z in the quadratic equation z² - 4z + 4 = 0. Quadratic equations are a fundamental topic in algebra, and mastering their solution is crucial for various mathematical applications. 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. The solutions to a quadratic equation, also known as roots or zeros, are the values of the variable (in this case, z) that satisfy the equation. These roots represent the points where the parabola represented by the quadratic equation intersects the x-axis. Solving quadratic equations is a foundational skill in algebra, with applications in various fields, including physics, engineering, economics, and computer science. This article aims to provide a step-by-step guide to solving the specific quadratic equation z² - 4z + 4 = 0, exploring different methods and highlighting key concepts. By the end of this discussion, you will be equipped with the knowledge and skills to tackle similar quadratic equations with confidence. Understanding the properties and solutions of quadratic equations is essential for advanced mathematical studies and real-world problem-solving. We will explore multiple methods for solving this equation, including factoring, completing the square, and using the quadratic formula. Each method offers a unique approach and provides valuable insights into the nature of quadratic equations. This article will not only guide you through the solution process but also enhance your understanding of the underlying mathematical principles. Let's embark on this journey to master the art of solving quadratic equations!

Method 1: Factoring the Quadratic Equation

One of the most efficient methods to solve quadratic equations is by factoring. Factoring involves expressing the quadratic expression as a product of two binomials. This method is particularly effective when the quadratic expression can be easily factored. In our case, the quadratic equation is z² - 4z + 4 = 0. To factor this equation, we look for two numbers that multiply to the constant term (4) and add up to the coefficient of the linear term (-4). These two numbers are -2 and -2, since (-2) * (-2) = 4 and (-2) + (-2) = -4. Therefore, we can rewrite the quadratic equation as (z - 2)(z - 2) = 0. This can be further simplified as (z - 2)² = 0. To find the values of z, we set each factor equal to zero. In this case, we have z - 2 = 0. Solving for z, we add 2 to both sides of the equation, which gives us z = 2. Since both factors are the same, we have a repeated root. This means that the quadratic equation has only one distinct solution. The factored form of the quadratic equation clearly demonstrates that z = 2 is the solution. Factoring is a powerful technique for solving quadratic equations, as it simplifies the problem into finding the roots of linear equations. It also provides insights into the structure and properties of quadratic expressions. By recognizing patterns and relationships between the coefficients and the roots, we can efficiently solve a wide range of quadratic equations. This method is not only applicable to simple quadratic equations but also serves as a foundation for solving more complex polynomial equations. Mastering factoring techniques is essential for any student of algebra, as it enhances problem-solving skills and deepens understanding of mathematical concepts. In the next sections, we will explore alternative methods for solving quadratic equations, further expanding our toolkit for tackling algebraic problems. Understanding factoring also helps in simplifying algebraic expressions and identifying special cases, such as perfect square trinomials and differences of squares.

Method 2: Using the Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation of the form ax² + bx + c = 0. This formula provides the solutions for x directly, regardless of whether the equation can be easily factored. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / (2a). In our specific equation, z² - 4z + 4 = 0, we can identify the coefficients as a = 1, b = -4, and c = 4. Plugging these values into the quadratic formula, we get: z = (-(-4) ± √((-4)² - 4(1)(4))) / (2(1)). Simplifying the expression inside the square root, we have: (-4)² - 4(1)(4) = 16 - 16 = 0. Thus, the quadratic formula becomes: z = (4 ± √0) / 2. Since the square root of 0 is 0, the equation simplifies to: z = (4 ± 0) / 2. This gives us: z = 4 / 2 = 2. Therefore, the quadratic formula yields a single solution, z = 2. This result aligns with our solution obtained through factoring. The quadratic formula is a powerful tool because it works for all quadratic equations, even those with complex or irrational roots. It is especially useful when factoring is difficult or not possible. The discriminant, b² - 4ac, plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has one real root (a repeated root); and if it is negative, the equation has two complex roots. In our case, the discriminant is 0, indicating a single real root. Mastering the quadratic formula is essential for anyone studying algebra and higher-level mathematics. It provides a reliable method for solving quadratic equations and forms the basis for understanding more advanced algebraic concepts. The quadratic formula is a cornerstone of algebra, enabling us to solve equations that arise in various mathematical and real-world contexts. Its versatility and applicability make it an indispensable tool in the mathematician's toolkit. Furthermore, understanding the derivation and application of the quadratic formula enhances our problem-solving abilities and critical thinking skills. The formula's elegant structure and its ability to provide solutions for any quadratic equation make it a testament to the power of mathematical abstraction and generalization.

Method 3: Completing the Square

Completing the square is another method for solving quadratic equations. This technique involves manipulating the quadratic equation into a perfect square trinomial form, which can then be easily solved. For the equation z² - 4z + 4 = 0, we can observe that the left side is already a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (az + b)² or (az - b)². In our case, z² - 4z + 4 can be written as (z - 2)². Therefore, the equation becomes (z - 2)² = 0. Taking the square root of both sides, we get: √((z - 2)²) = √0. This simplifies to z - 2 = 0. Adding 2 to both sides of the equation, we find: z = 2. Thus, completing the square also yields the solution z = 2. This method is particularly useful for understanding the structure of quadratic equations and their relationship to perfect square trinomials. Completing the square is a valuable technique because it can be applied to any quadratic equation, even those that are not easily factored. It also provides a deeper understanding of the relationship between the coefficients and the roots of the equation. The process of completing the square involves adding and subtracting a constant term to create a perfect square trinomial. This constant term is determined by taking half of the coefficient of the linear term and squaring it. In general, for an equation of the form ax² + bx + c = 0, we can divide through by a (if a ≠ 0) and then add and subtract (b/2a)² to complete the square. The completed square form allows us to easily identify the vertex of the parabola represented by the quadratic equation, which is a crucial concept in graphing quadratic functions. Completing the square not only helps in solving quadratic equations but also provides valuable insights into the geometric interpretation of quadratic functions. This method bridges the gap between algebraic manipulation and graphical representation, enhancing our overall understanding of quadratic equations. Furthermore, completing the square is a foundational technique for deriving the quadratic formula itself, highlighting its significance in the broader context of algebra. By mastering completing the square, we gain a deeper appreciation for the interconnectedness of various algebraic concepts and techniques.

Conclusion: The Solution and Its Significance

In summary, we have explored three different methods—factoring, using the quadratic formula, and completing the square—to solve the quadratic equation z² - 4z + 4 = 0. All three methods consistently yield the same solution: z = 2. This consistent result reinforces the validity of each method and demonstrates the interconnectedness of different algebraic techniques. The solution z = 2 is a repeated root, indicating that the parabola represented by the quadratic equation touches the x-axis at only one point. This point is also the vertex of the parabola, which in this case is (2, 0). Understanding the nature of the roots of a quadratic equation is crucial in various applications, including graphing, optimization problems, and modeling real-world phenomena. The fact that the equation has a repeated root also implies that the discriminant (b² - 4ac) is equal to zero, which we confirmed when using the quadratic formula. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications in diverse fields such as physics, engineering, economics, and computer science. Whether we are designing bridges, modeling financial markets, or developing algorithms, quadratic equations often play a central role. The methods we have discussed provide a comprehensive toolkit for solving quadratic equations, each with its own strengths and applications. Factoring is efficient for equations with easily factorable expressions, the quadratic formula is universally applicable, and completing the square provides a deeper understanding of the equation's structure. By mastering these techniques, we can confidently tackle a wide range of quadratic equation problems. Furthermore, the process of solving quadratic equations enhances our problem-solving skills, critical thinking abilities, and overall mathematical maturity. The consistent solution obtained through different methods underscores the importance of mathematical rigor and the beauty of mathematical consistency. As we continue our mathematical journey, the skills and insights gained from solving quadratic equations will serve as a solid foundation for tackling more complex problems and exploring advanced concepts. The solution z = 2 is not just a numerical answer; it is a key to understanding the behavior and properties of the quadratic equation and its graphical representation.

Final Answer: The final answer is (A) z=2