Solving The Quadratic Equation 4x² + 40x + 100 = 0 A Step-by-Step Guide

Introduction: Understanding Quadratic Equations

Quadratic equations are fundamental concepts in algebra, playing a crucial role in various mathematical and real-world applications. These equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, can model a wide array of phenomena, from the trajectory of a projectile to the design of curved structures. In this comprehensive guide, we will delve into the specifics of solving the quadratic equation 4x² + 40x + 100 = 0, exploring different methods and providing a step-by-step approach to arrive at the solution. Understanding how to solve quadratic equations is not only essential for mathematical proficiency but also provides valuable problem-solving skills applicable across various disciplines. Whether you are a student grappling with algebra, an engineer designing systems, or simply a curious mind, mastering quadratic equations opens doors to a deeper understanding of the world around us. We will break down each method, explain the underlying principles, and provide examples to ensure a clear and thorough understanding.

Method 1: Factoring the Quadratic Equation

Factoring is a powerful method for solving quadratic equations, particularly when the equation can be expressed as a product of two binomials. This technique relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. To effectively apply factoring, we first need to identify two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. In the equation 4x² + 40x + 100 = 0, we can simplify the process by first dividing the entire equation by the common factor of 4, resulting in x² + 10x + 25 = 0. This simplified equation is easier to factor. Next, we look for two numbers that multiply to the constant term (25) and add up to the coefficient of the linear term (10). These numbers are 5 and 5, since 5 * 5 = 25 and 5 + 5 = 10. Therefore, we can rewrite the quadratic equation as (x + 5)(x + 5) = 0, or (x + 5)² = 0. Setting each factor equal to zero, we get x + 5 = 0, which yields the solution x = -5. This solution is a repeated root, meaning that the quadratic equation has only one distinct solution. Factoring is an efficient method when applicable, but it may not always be straightforward for all quadratic equations, especially those with non-integer roots or complex coefficients. In such cases, alternative methods like the quadratic formula or completing the square become more suitable.

Method 2: Using the Quadratic Formula

The quadratic formula is a versatile and universally applicable method for solving quadratic equations of the form ax² + bx + c = 0. This formula provides a direct solution for x regardless of the nature of the roots (real, complex, rational, or irrational). The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, 4x² + 40x + 100 = 0, we identify the coefficients as a = 4, b = 40, and c = 100. Plugging these values into the quadratic formula, we get:

x = (-40 ± √(40² - 4 * 4 * 100)) / (2 * 4)

Simplifying the expression under the square root, we have:

x = (-40 ± √(1600 - 1600)) / 8

The discriminant (the term under the square root), b² - 4ac, is 0, indicating that the equation has one real, repeated root. Continuing the simplification:

x = (-40 ± √0) / 8 x = -40 / 8 x = -5

Thus, the quadratic formula confirms that the solution to the equation 4x² + 40x + 100 = 0 is x = -5. The quadratic formula is particularly useful when factoring is difficult or impossible, as it provides a systematic approach to finding the roots of any quadratic equation. It is an indispensable tool in algebra and is widely used in various fields of mathematics, science, and engineering.

Method 3: Completing the Square

Completing the square is another powerful technique for solving quadratic equations, providing a systematic way to rewrite the equation in a form that allows for easy isolation of the variable x. This method involves transforming the quadratic equation ax² + bx + c = 0 into the form (x + p)² = q, where p and q are constants. Once in this form, the solution can be found by taking the square root of both sides and solving for x. To apply completing the square to the equation 4x² + 40x + 100 = 0, we first divide the equation by the leading coefficient, 4, to obtain x² + 10x + 25 = 0. Next, we focus on the terms x² + 10x. To complete the square, we need to add and subtract (b/2)², where b is the coefficient of the x term. In this case, b = 10, so (b/2)² = (10/2)² = 25. Adding and subtracting 25 within the equation doesn't change its value, but it allows us to rewrite it in the desired form. The equation can be rewritten as x² + 10x + 25 - 25 + 25 = 0. The first three terms, x² + 10x + 25, form a perfect square trinomial, which can be factored as (x + 5)². The equation now becomes (x + 5)² = 0. Taking the square root of both sides, we get x + 5 = 0, which leads to the solution x = -5. Completing the square is not only a method for solving quadratic equations but also a fundamental technique used in various mathematical contexts, such as deriving the quadratic formula and analyzing conic sections. It provides a deep understanding of the structure of quadratic equations and their solutions.

Conclusion: Summarizing the Solution

In conclusion, we have explored three distinct methods for solving the quadratic equation 4x² + 40x + 100 = 0: factoring, using the quadratic formula, and completing the square. Each method provides a unique approach to finding the solution, but they all converge on the same result: x = -5. Factoring, when applicable, offers a straightforward and intuitive way to solve quadratic equations by expressing them as a product of binomials. The quadratic formula, on the other hand, is a universally applicable method that provides a direct solution regardless of the nature of the roots. Completing the square not only solves the equation but also offers insights into the structure of quadratic expressions and their transformations. The equation 4x² + 40x + 100 = 0 serves as an excellent example for illustrating these methods due to its perfect square trinomial nature, which simplifies the factoring and completing the square processes. Understanding these methods equips you with a comprehensive toolkit for tackling quadratic equations in various mathematical and real-world scenarios. Whether you prefer the elegance of factoring, the generality of the quadratic formula, or the structural insights of completing the square, mastering these techniques will enhance your problem-solving skills and deepen your understanding of algebra. The solution x = -5 represents a repeated root, indicating that the parabola represented by the equation touches the x-axis at a single point, highlighting the unique characteristics of quadratic equations with repeated roots.