Solving Quadratic Equation X² - 10x + 25 = 0 A Comprehensive Guide

Introduction to Quadratic Equations

In the realm of mathematics, quadratic equations hold a significant place. Understanding how to solve them is a fundamental skill, essential not only for academic pursuits but also for various real-world 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 is not equal to zero. These equations can model a wide range of phenomena, from the trajectory of a projectile to the calculation of areas and optimization problems. Mastering the techniques to solve quadratic equations opens doors to more advanced mathematical concepts and practical problem-solving scenarios.

When tackling quadratic equations, there are several methods available, each with its own strengths and applicability. Among the most common methods are factoring, completing the square, and using the quadratic formula. Factoring involves breaking down the quadratic expression into two linear expressions, which can then be solved individually. Completing the square is a technique that transforms the equation into a perfect square trinomial, making it easier to solve. The quadratic formula is a universal method that can solve any quadratic equation, regardless of its complexity. Choosing the appropriate method depends on the specific equation and personal preference. However, understanding all these methods provides a comprehensive toolkit for solving any quadratic equation that arises.

The equation we will address today, x² - 10x + 25 = 0, presents a classic example of a quadratic equation that can be solved using multiple methods. This specific equation is particularly interesting because it is a perfect square trinomial, which makes factoring a straightforward and efficient approach. However, we will also explore alternative methods to provide a thorough understanding of how different techniques can be applied to the same problem. By examining various solution pathways, we can gain a deeper appreciation for the versatility and interconnectedness of mathematical problem-solving strategies. This comprehensive approach will not only help in solving this particular equation but also equip us with the skills to tackle a broader range of quadratic equations.

Method 1: Factoring

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily expressed as a product of two binomials. This method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. In the case of x² - 10x + 25 = 0, we recognize that the quadratic expression is a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. Identifying such patterns significantly simplifies the factoring process and provides a quick route to the solution.

To factor x² - 10x + 25, we look for two numbers that multiply to 25 (the constant term) and add up to -10 (the coefficient of the x term). These numbers are -5 and -5. Therefore, the quadratic expression can be factored as (x - 5)(x - 5), which can also be written as (x - 5)². Setting this equal to zero gives us (x - 5)² = 0. This equation implies that x - 5 must be equal to zero, since the square of any non-zero number cannot be zero. Thus, we have a single, repeated factor, which leads to a single solution for x.

Solving x - 5 = 0 is a simple algebraic step. By adding 5 to both sides of the equation, we isolate x and find that x = 5. This is the only solution to the quadratic equation x² - 10x + 25 = 0. The fact that we have a repeated factor indicates that the quadratic equation has a single, real root, which means the parabola represented by the equation touches the x-axis at only one point. This solution highlights the elegance and efficiency of factoring when dealing with perfect square trinomials. Factoring not only provides the solution but also gives insights into the nature of the roots of the quadratic equation.

Method 2: Completing the Square

Completing the square is a versatile method for solving quadratic equations, especially useful when factoring is not immediately obvious or when the quadratic expression is not easily factorable. This technique involves transforming the quadratic equation into a form where one side is a perfect square trinomial. The method is based on the algebraic identity (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b². By manipulating the equation to match one of these forms, we can easily extract the roots.

To apply completing the square to the equation x² - 10x + 25 = 0, we first ensure that the coefficient of x² is 1, which it already is in this case. Next, we focus on the x term (-10x). We take half of the coefficient of x (-10/2 = -5) and square it ((-5)² = 25). Notice that the constant term in our equation is already 25, which means the given equation is already a perfect square trinomial. This simplifies the process significantly, as no additional steps are needed to complete the square.

Since x² - 10x + 25 is a perfect square trinomial, it can be written as (x - 5)². Thus, the equation becomes (x - 5)² = 0. Taking the square root of both sides, we get x - 5 = 0. Solving for x, we add 5 to both sides, which gives us x = 5. This result confirms our solution obtained through factoring and demonstrates the effectiveness of the completing the square method. Completing the square provides a systematic approach to solving quadratic equations, even when the equation is not initially in a convenient form. This method is particularly valuable as it leads to a general understanding of quadratic equations and their solutions.

Method 3: Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. It provides a direct way to find the roots, regardless of whether the equation is factorable or easily amenable to completing the square. The formula is derived from the process of completing the square and is given by:

x = (-b ± √(b² - 4ac)) / (2a)

This formula can be applied to any quadratic equation by simply substituting the coefficients a, b, and c into the formula and simplifying the expression. The term b² - 4ac under the square root is known as the discriminant, which provides valuable information about the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots.

For the equation x² - 10x + 25 = 0, we identify the coefficients as a = 1, b = -10, and c = 25. Substituting these values into the quadratic formula, we get:

x = (-(-10) ± √((-10)² - 4 * 1 * 25)) / (2 * 1)

Simplifying the expression:

x = (10 ± √(100 - 100)) / 2

x = (10 ± √0) / 2

x = 10 / 2

x = 5

The result shows that there is one real root, x = 5. This aligns with our findings from the factoring and completing the square methods. The discriminant being zero confirms that there is a repeated root. The quadratic formula provides a reliable and straightforward method for solving quadratic equations, particularly when other methods may be cumbersome. Its generality makes it an indispensable tool in mathematics and related fields.

Conclusion

In summary, we have successfully solved the quadratic equation x² - 10x + 25 = 0 using three distinct methods: factoring, completing the square, and the quadratic formula. Each method provided the same solution, x = 5, reinforcing the consistency and reliability of these techniques. The equation's structure as a perfect square trinomial made factoring a particularly efficient approach, while completing the square demonstrated a systematic method applicable to a broader range of quadratic equations. The quadratic formula served as a universal tool, confirming the solution and illustrating its broad applicability.

Understanding these different methods not only enhances our ability to solve quadratic equations but also deepens our appreciation for the interconnectedness of mathematical concepts. Each method offers a unique perspective and can be advantageous depending on the specific equation and the problem-solver's preferences. Factoring is often the quickest method when applicable, but it requires recognizing patterns. Completing the square provides a step-by-step approach that is particularly useful for understanding the structure of quadratic equations. The quadratic formula is a reliable tool that can be applied to any quadratic equation, regardless of its complexity.

By exploring multiple solution pathways, we have gained a more robust understanding of quadratic equations and the techniques used to solve them. This comprehensive approach not only prepares us for more advanced mathematical studies but also equips us with valuable problem-solving skills that can be applied in various contexts. The ability to solve quadratic equations is a fundamental skill in mathematics, and mastering these methods lays a strong foundation for further mathematical explorations. Whether you prefer the elegance of factoring, the systematic approach of completing the square, or the generality of the quadratic formula, having these tools at your disposal will undoubtedly enhance your mathematical prowess.