Solving -x^2 + 4x = X - 4 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of finding the solutions to the quadratic equation $-x^2 + 4x = x - 4$. Quadratic equations are a fundamental topic in algebra, and understanding how to solve them is crucial for various mathematical applications. We will explore the steps involved in rearranging the equation into standard form, and then we will apply different methods to find the roots or solutions. This detailed explanation will not only provide the answer but also enhance your understanding of quadratic equations and problem-solving techniques.

Solving quadratic equations is a core skill in mathematics, with applications spanning across diverse fields such as physics, engineering, and computer science. The ability to manipulate and solve these equations is essential for modeling real-world phenomena and making accurate predictions. In this article, we will focus on the given equation, $-x^2 + 4x = x - 4$, and systematically break down the solution process. Our goal is to not only find the correct answer but also to illustrate the underlying principles and methods applicable to similar problems. This step-by-step approach will help you build confidence in your algebraic skills and improve your ability to tackle more complex mathematical challenges. Let's begin by understanding the initial setup and the need for rearranging the equation into a more manageable form.

The importance of understanding quadratic equations cannot be overstated. These equations appear in numerous mathematical contexts and are the building blocks for more advanced concepts. By mastering the techniques for solving them, you are laying a strong foundation for your mathematical journey. This article serves as a comprehensive guide, ensuring that you not only arrive at the correct solution but also grasp the reasoning behind each step. We will explore the equation $-x^2 + 4x = x - 4$ in depth, providing a clear and methodical approach to finding its solutions. By the end of this article, you will have a solid understanding of how to handle similar quadratic equations and a deeper appreciation for the elegance and power of algebraic methods. Let's embark on this mathematical exploration together, uncovering the solutions and the underlying principles that make them possible.

Step 1: Rearranging the Equation

The first step in solving the equation $-x^2 + 4x = x - 4$ is to rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. This form allows us to easily identify the coefficients $a$, $b$, and $c$, which are essential for applying methods such as factoring, completing the square, or using the quadratic formula. To rearrange the equation, we need to move all terms to one side, leaving zero on the other side. This involves performing algebraic operations that maintain the equality of the equation.

To begin, we can subtract $x$ from both sides of the equation to eliminate the $x$ term on the right side. This gives us $-x^2 + 4x - x = -4$. Simplifying the left side by combining like terms, we have $-x^2 + 3x = -4$. Next, we want to move the constant term from the right side to the left side. To do this, we add $4$ to both sides, resulting in $-x^2 + 3x + 4 = 0$. Now the equation is in the standard quadratic form, but it's generally preferred to have the leading coefficient (the coefficient of $x^2$) as a positive number. To achieve this, we can multiply the entire equation by $-1$. This changes the sign of each term, giving us $x^2 - 3x - 4 = 0$. We now have the quadratic equation in its standard form, making it ready for the next steps in the solution process.

Rearranging the equation into standard form is a critical step because it sets the stage for applying various solution methods. The standard form, $ax^2 + bx + c = 0$, provides a clear structure that allows us to identify the coefficients $a$, $b$, and $c$, which are crucial for both factoring and using the quadratic formula. This methodical approach ensures that we are working with a consistent and manageable format, reducing the likelihood of errors in subsequent steps. By transforming the original equation, $-x^2 + 4x = x - 4$, into $x^2 - 3x - 4 = 0$, we have created a clear pathway to finding the solutions. This initial rearrangement highlights the importance of algebraic manipulation in simplifying complex equations and making them solvable.

This step of transforming the equation is not just a mechanical process; it is a fundamental aspect of problem-solving in algebra. By understanding how to rearrange equations, you gain the ability to manipulate mathematical expressions to suit your needs. This skill is invaluable not only for solving quadratic equations but also for tackling a wide range of algebraic problems. The process of moving terms, combining like terms, and ensuring the leading coefficient is positive demonstrates the power of algebraic manipulation in simplifying complex problems. With the equation now in the standard form $x^2 - 3x - 4 = 0$, we are well-positioned to move forward and explore the methods for finding its solutions. The next steps will involve either factoring the quadratic expression or applying the quadratic formula, both of which are powerful techniques for solving these types of equations.

Step 2: Factoring the Quadratic

Now that we have the quadratic equation in the standard form $x^2 - 3x - 4 = 0$, we can attempt to solve it by factoring. Factoring is a method that involves expressing the quadratic expression as a product of two binomials. If we can successfully factor the expression, we can then set each factor equal to zero and solve for $x$. This method relies on finding two numbers that multiply to the constant term ($-4$ in this case) and add up to the coefficient of the $x$ term ($-3$ in this case).

To factor the quadratic $x^2 - 3x - 4$, we look for two numbers that multiply to $-4$ and add to $-3$. By considering the factors of $-4$, we can identify the pair $-4$ and $1$. These numbers satisfy our conditions: $(-4) imes 1 = -4$ and $(-4) + 1 = -3$. Therefore, we can rewrite the quadratic expression as $(x - 4)(x + 1)$. Now we have factored the equation into the form $(x - 4)(x + 1) = 0$. To find the solutions, we set each factor equal to zero: $x - 4 = 0$ and $x + 1 = 0$.

Factoring the quadratic is an efficient method for solving quadratic equations when the expression can be easily factored. This technique relies on the ability to recognize patterns and find the right combination of factors that satisfy the conditions. In our case, identifying the numbers $-4$ and $1$ was the key to successfully factoring the quadratic expression $x^2 - 3x - 4$. The process of breaking down the quadratic into two binomials, $(x - 4)(x + 1)$, allows us to transform a single equation into two simpler equations, making the solutions more accessible. Factoring is not only a practical method but also a valuable exercise in algebraic thinking, reinforcing the relationships between the coefficients and the roots of the equation. By mastering factoring, you gain a powerful tool for solving quadratic equations and a deeper understanding of the structure of these expressions.

The ability to identify the correct factors is a critical skill in algebra. It requires careful consideration of the numbers that multiply to the constant term and add up to the coefficient of the linear term. In this instance, the factors $-4$ and $1$ perfectly fit the requirements for factoring the quadratic $x^2 - 3x - 4$. Once the expression is factored, the process of finding the solutions becomes straightforward. Setting each factor equal to zero and solving for $x$ is a direct application of the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This principle is fundamental to solving equations by factoring and provides a clear pathway to the solutions. With the quadratic expression now factored as $(x - 4)(x + 1) = 0$, we are ready to determine the values of $x$ that satisfy the equation.

Step 3: Solving for x

After factoring the quadratic equation into $(x - 4)(x + 1) = 0$, we now proceed to solve for $x$. To do this, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property allows us to break the single equation into two simpler equations, each of which can be solved independently.

We set each factor equal to zero: $x - 4 = 0$ and $x + 1 = 0$. For the first equation, $x - 4 = 0$, we add $4$ to both sides to isolate $x$, which gives us $x = 4$. For the second equation, $x + 1 = 0$, we subtract $1$ from both sides to isolate $x$, resulting in $x = -1$. Thus, the solutions to the quadratic equation are $x = 4$ and $x = -1$.

Solving for x involves applying basic algebraic principles to isolate the variable in each of the equations derived from the factors. The zero-product property is the cornerstone of this step, providing a clear and logical method for finding the values of $x$ that make the equation true. By setting each factor equal to zero, we create two linear equations that are easily solved using simple addition or subtraction. The solutions $x = 4$ and $x = -1$ represent the points where the quadratic function intersects the x-axis, and they are the roots of the equation. This process highlights the power of factoring as a technique for simplifying complex equations and finding their solutions in a straightforward manner. With these solutions in hand, we have successfully navigated the quadratic equation and determined the values of $x$ that satisfy the original problem.

This step of isolating the variable x is a fundamental skill in algebra. It demonstrates the importance of performing inverse operations to solve equations. By adding $4$ to both sides of $x - 4 = 0$ and subtracting $1$ from both sides of $x + 1 = 0$, we effectively isolate $x$ and determine its value in each case. The resulting solutions, $x = 4$ and $x = -1$, are the answers to the original quadratic equation. These values can be verified by substituting them back into the original equation to ensure they satisfy the equality. The process of solving for $x$ reinforces the principles of algebraic manipulation and provides a concrete outcome to the factoring process. With the solutions now identified, we can confidently conclude our analysis of the quadratic equation and move on to understanding the broader implications of these solutions.

Conclusion

In conclusion, the solutions to the quadratic equation $-x^2 + 4x = x - 4$ are $x = -1$ and $x = 4$. We arrived at these solutions by first rearranging the equation into the standard quadratic form, $x^2 - 3x - 4 = 0$, then factoring the quadratic expression into $(x - 4)(x + 1) = 0$, and finally, applying the zero-product property to solve for $x$. This step-by-step process demonstrates a clear and methodical approach to solving quadratic equations.

By summarizing the steps, we can see the logical progression from the initial equation to the final solutions. Rearranging the equation into standard form is a critical first step, as it sets the stage for applying factoring or the quadratic formula. Factoring, when possible, provides an efficient way to break down the quadratic expression into simpler terms. The zero-product property then allows us to find the values of $x$ that make each factor equal to zero, thus solving the equation. This process not only provides the solutions but also reinforces the underlying principles of algebra. The solutions $x = -1$ and $x = 4$ represent the points where the quadratic function intersects the x-axis, and understanding how to find these points is essential for various mathematical applications.

This exploration of solving quadratic equations highlights the importance of algebraic manipulation and problem-solving techniques. Quadratic equations are a fundamental topic in mathematics, and mastering the methods for solving them is crucial for success in higher-level courses. The process of rearranging, factoring, and applying the zero-product property demonstrates the power of algebra in simplifying complex problems. By following a structured approach and understanding the underlying principles, you can confidently tackle quadratic equations and other algebraic challenges. The solutions $x = -1$ and $x = 4$ are not just numbers; they are the result of a systematic and logical process that underscores the beauty and power of mathematical reasoning.