Solve U² + 47u = 0: A Step-by-Step Guide

Unveiling the Solutions to u² + 47u = 0: A Comprehensive Guide

In the realm of mathematics, solving equations is a fundamental skill. Quadratic equations, characterized by the presence of a squared term, often present a unique challenge. In this comprehensive guide, we will embark on a journey to dissect the equation u² + 47u = 0 and extract its solutions with clarity and precision. Our exploration will encompass a step-by-step approach, ensuring that every nuance is meticulously addressed. This exploration will not only illuminate the solution but also enhance your understanding of quadratic equations. We will delve into the underlying principles that govern their behavior and equip you with the tools to tackle similar problems with confidence. This journey is not merely about finding the answer; it's about cultivating a deeper appreciation for the elegance and logic that permeate the world of mathematics.

Factoring: The Key to Unlocking the Solutions

At the heart of our strategy lies the principle of factoring. Factoring involves expressing a mathematical expression as a product of its constituent factors. This technique proves particularly potent when dealing with quadratic equations, as it allows us to transform the equation into a form that is amenable to direct solution. In the given equation, u² + 47u = 0, we observe a common factor lurking within both terms: the variable 'u'. By skillfully extracting this common factor, we can rewrite the equation in a more revealing form. The act of factoring is akin to unveiling the hidden structure of the equation, exposing the fundamental building blocks that govern its behavior. It is a testament to the power of mathematical manipulation, allowing us to transform complex expressions into simpler, more manageable forms. This transformation is not merely aesthetic; it is a strategic maneuver that paves the way for a straightforward solution.

Extracting the Common Factor: A Step-by-Step Approach

To embark on the process of factoring, we meticulously identify the common factor shared by the terms in the equation. In the equation u² + 47u = 0, the common factor is readily apparent: the variable 'u'. With the common factor identified, we proceed to extract it from each term. This involves dividing each term by the common factor and expressing the result within parentheses. The process of extraction is akin to dissecting the equation, isolating the common element that binds the terms together. It is a systematic approach that ensures no element is overlooked, paving the way for an accurate and complete factorization. By extracting 'u' from u², we obtain 'u', and by extracting 'u' from 47u, we obtain 47. This careful extraction transforms the equation into the factored form: u(u + 47) = 0. This factored form is not merely a cosmetic change; it is a pivotal step that unlocks the solutions to the equation.

The Zero Product Property: A Cornerstone of Solution Finding

The factored form of the equation, u(u + 47) = 0, holds the key to unraveling the solutions. Here, we invoke the Zero Product Property, a fundamental principle in algebra. This property asserts that if the product of two or more factors is equal to zero, then at least one of the factors must be zero. In the context of our equation, this property implies that either 'u' must be zero or '(u + 47)' must be zero. The Zero Product Property is a cornerstone of solution finding, providing a direct link between the factored form of an equation and its solutions. It is a powerful tool that transforms the task of solving an equation into a series of simpler tasks: setting each factor equal to zero and solving the resulting equations. This property is not merely a mathematical trick; it is a reflection of the fundamental nature of multiplication and the role of zero as an absorbing element.

Setting Each Factor to Zero: Unveiling the Potential Solutions

Guided by the Zero Product Property, we proceed to set each factor in the factored equation, u(u + 47) = 0, equal to zero. This yields two distinct equations: u = 0 and u + 47 = 0. Each of these equations represents a potential solution to the original quadratic equation. The act of setting each factor to zero is akin to exploring different avenues, each leading to a possible solution. It is a systematic approach that ensures no potential solution is overlooked, providing a comprehensive exploration of the solution space. These equations are not merely mathematical statements; they are clues that guide us towards the ultimate answer.

Solving the Linear Equations: Extracting the Solutions

With the factors set to zero, we now confront two simple linear equations: u = 0 and u + 47 = 0. The equation u = 0 is already in its solved form, directly revealing one of the solutions. To solve the equation u + 47 = 0, we employ the principle of isolating the variable. This involves subtracting 47 from both sides of the equation, resulting in u = -47. The process of solving linear equations is a fundamental skill in algebra, involving the manipulation of equations to isolate the unknown variable. It is a testament to the power of algebraic manipulation, allowing us to transform equations into forms that directly reveal the solutions. With the two linear equations solved, we have successfully extracted the solutions to the original quadratic equation.

The Solutions Unveiled: u = 0, -47

Having meticulously navigated the steps of factoring, applying the Zero Product Property, and solving the resulting linear equations, we arrive at the solutions to the equation u² + 47u = 0. The solutions are u = 0 and u = -47. These solutions represent the values of 'u' that satisfy the original equation, making it a true statement. The solutions are not merely numbers; they are the culmination of a mathematical journey, a testament to the power of logical deduction and algebraic manipulation. These solutions stand as a complete and accurate answer to the problem, satisfying the given conditions and constraints.

Expressing the Solutions: Integers in Simplest Form

The problem statement mandates that the solutions be expressed as integers, proper fractions, or improper fractions in simplest form. In this case, both solutions, u = 0 and u = -47, are integers. Moreover, they are already expressed in their simplest form. The requirement to express solutions in a specific form underscores the importance of mathematical precision and clarity. It is a reminder that solutions are not merely numerical values; they are mathematical statements that must adhere to specific conventions and standards. The fact that the solutions are integers in simplest form is a testament to the elegance and conciseness of the answer.

Therefore, the solutions to the equation u² + 47u = 0 are:

u = 0, -47

This detailed exploration has not only unveiled the solutions but also illuminated the underlying principles and techniques involved in solving quadratic equations. By mastering these concepts, you can confidently tackle a wide range of mathematical challenges.

Keywords

Keywords: quadratic equations, factoring, Zero Product Property, solutions, integers, simplest form, algebraic manipulation, solving equations, mathematical principles, equation.