Solving $ \sqrt{-2x+4} - X = 2 $ A Step-by-Step Guide

Jul 23, 2025 by ADMIN 56 views

$Solving the Equation ${ \sqrt{-2x+4} - x = 2 }$ Simultaneously: A Comprehensive Guide$

Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving the equation ${ \sqrt{-2x+4} - x = 2 }$ simultaneously. We will explore the step-by-step method, the underlying concepts, and potential pitfalls to avoid. Understanding how to solve equations like this is crucial for various mathematical applications, making this a valuable skill for students and professionals alike. The key to successfully solving this equation lies in isolating the square root, squaring both sides, and carefully checking for extraneous solutions. By following this guide, you will gain a solid understanding of the techniques involved and be able to apply them to similar problems.

Understanding the Equation

To begin, let's break down the equation ${ \sqrt{-2x+4} - x = 2 }$ . This equation involves a square root, which adds a layer of complexity. Our goal is to find the value(s) of x that satisfy this equation. The presence of the square root means we need to be cautious about potential extraneous solutions, which are solutions that arise during the solving process but do not actually satisfy the original equation. Extraneous solutions often occur when squaring both sides of an equation, as this operation can introduce solutions that are not valid in the original context. Therefore, a crucial step in solving this type of equation is to always check the solutions we obtain by substituting them back into the original equation. Understanding the domain of the square root function is also crucial. The expression inside the square root, ${-2x+4}$ , must be non-negative. This gives us a constraint on the possible values of x. Solving the inequality ${-2x+4 \geq 0}$ will help us determine the valid range of x values we should consider. In this case, we find that x must be less than or equal to 2. This is an important piece of information that will help us filter out any extraneous solutions we might find later. By understanding these aspects of the equation, we set the stage for a systematic approach to finding the solution.

Step-by-Step Solution

Isolate the Square Root: The first step is to isolate the square root term. We can do this by adding x to both sides of the equation:

${ \sqrt{-2x+4} = x + 2 }$

Isolating the square root is a critical first step because it allows us to eliminate the square root by squaring both sides in the next step. This simplifies the equation and allows us to work with a more manageable expression. Without isolating the square root, squaring both sides would lead to a more complex equation with both square root terms and linear terms, making it harder to solve. By isolating the square root, we set ourselves up for a cleaner algebraic manipulation.
Square Both Sides: Now, we square both sides of the equation to eliminate the square root:

${ (\sqrt{-2x+4})^2 = (x + 2)^2 }$

${ -2x + 4 = x^2 + 4x + 4 }$

Squaring both sides of the equation is a fundamental technique for dealing with square roots in equations. It allows us to transform the equation into a more familiar form, typically a polynomial equation, which we can then solve using standard algebraic methods. However, it's crucially important to remember that squaring both sides can introduce extraneous solutions. This is because the squaring operation can make two unequal quantities appear equal (for example, squaring both 2 and -2 results in 4). This is why checking the solutions we obtain against the original equation is a mandatory step.
Rearrange into a Quadratic Equation: Next, we rearrange the equation into a standard quadratic form:

${ 0 = x^2 + 6x }$

Rearranging the equation into the standard quadratic form ${ax^2 + bx + c = 0}$ is a key step in solving for x. This form allows us to readily apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, we've moved all terms to one side of the equation to set it equal to zero, which is the standard form for a quadratic equation. This rearrangement simplifies the process of finding the solutions, as it allows us to leverage well-established techniques for solving quadratic equations. By getting the equation into this standard form, we make it easier to identify the coefficients and apply the appropriate solution method.
Solve the Quadratic Equation: We can factor the quadratic equation:

${ 0 = x(x + 6) }$

This gives us two potential solutions:

${ x = 0 \text{ or } x = -6 }$

Factoring is a powerful technique for solving quadratic equations, and it's often the quickest method when the quadratic expression can be easily factored. By factoring the quadratic expression into the product of two linear factors, we can set each factor equal to zero and solve for x. This gives us two potential solutions. However, it's essential to remember that these are just potential solutions at this stage. We need to verify whether they actually satisfy the original equation, as squaring both sides can introduce extraneous solutions. This step highlights the importance of checking solutions in the original equation to ensure their validity.
Check for Extraneous Solutions: We must check both potential solutions in the original equation:

For x = 0:

${ \sqrt{-2(0) + 4} - 0 = \sqrt{4} = 2 }$

This solution is valid.

For x = -6:

${ \sqrt{-2(-6) + 4} - (-6) = \sqrt{16} + 6 = 4 + 6 = 10 \neq 2 }$

This solution is extraneous.

Checking for extraneous solutions is a critical step when solving equations involving square roots. As mentioned earlier, squaring both sides of an equation can introduce solutions that do not satisfy the original equation. This occurs because the squaring operation eliminates the sign of a term, potentially making two unequal quantities appear equal. In this case, we found two potential solutions, 0 and -6. By substituting each of these values back into the original equation, we discovered that x = 0 is a valid solution, while x = -6 is an extraneous solution. This demonstrates the necessity of this verification step to ensure that the solutions we obtain are genuine solutions to the original equation.

Conclusion

The solution to the equation ${ \sqrt{-2x+4} - x = 2 }$ is x = 0. This process highlights the importance of isolating the square root, squaring both sides, solving the resulting equation, and, most importantly, checking for extraneous solutions. By mastering these steps, you can confidently tackle similar equations in mathematics. Understanding the underlying principles and potential pitfalls is key to success in solving equations. Remember to always be mindful of extraneous solutions when dealing with square roots, and to meticulously check your answers against the original equation. This will ensure that you arrive at the correct and valid solutions.