Rewriting Equations A Guide To Transforming Sqrt(x + 4) = X Into X + 4 = X^2

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In the realm of algebra, solving equations often involves manipulating them into different forms while preserving their fundamental meaning. This process allows us to isolate variables, simplify expressions, and ultimately find the solutions that satisfy the equation. In this comprehensive guide, we will delve into the process of identifying which equation can be rewritten as x + 4 = x², assuming x > 0. This problem requires a solid understanding of algebraic manipulation, particularly dealing with square roots and quadratic equations. We will explore various techniques to transform equations and determine their equivalence. By the end of this guide, you will not only be able to solve this specific problem but also gain a deeper understanding of how to manipulate algebraic equations effectively. Understanding the core concepts of algebraic manipulation is crucial. This involves recognizing how to apply operations such as squaring, adding, subtracting, multiplying, and dividing both sides of an equation without altering its solutions. These skills are fundamental in simplifying equations and transforming them into a more manageable form. Our focus will be on equations involving square roots, which require careful handling to avoid introducing extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original one, often arising from squaring both sides of an equation. The target equation, x + 4 = x², is a quadratic equation. Quadratic equations are of the form ax² + bx + c = 0, where a, b, and c are constants. Solving quadratic equations typically involves factoring, completing the square, or using the quadratic formula. In this context, we need to determine which of the given equations can be manipulated into this quadratic form. We will examine each option systematically, applying algebraic transformations to see if we can arrive at the target equation. The ability to recognize patterns and apply appropriate techniques is key to solving such problems efficiently. This guide aims to provide a clear and detailed explanation of each step, ensuring that you grasp the underlying principles and can apply them to similar problems in the future.

Understanding the Target Equation: x + 4 = x²

Before we dive into the given options, let's take a closer look at our target equation: x + 4 = x². This is a quadratic equation, which can be rearranged into the standard form ax² + bx + c = 0. To do this, we subtract x and 4 from both sides, resulting in: x² - x - 4 = 0. This standard form is crucial because it allows us to apply various methods for solving quadratic equations, such as the quadratic formula or completing the square. However, our primary goal here is not to solve the equation but to identify which of the given options can be transformed into this form. Understanding the structure of the target equation is essential for recognizing equivalent forms. The equation involves a squared term (x²), a linear term (x), and a constant term (-4). Any equation that can be manipulated to match this structure is a potential match. One common technique for dealing with equations involving square roots is to square both sides. However, it's important to remember that squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, we need to be cautious and verify our solutions in the original equation. In the context of this problem, squaring both sides might be a necessary step to eliminate the square root and obtain a quadratic equation. We will carefully consider the implications of squaring and ensure that any potential solutions are checked for validity. The condition x > 0 is also significant. It restricts our attention to positive solutions, which can simplify the process of verifying solutions. Negative solutions, even if they satisfy the quadratic equation, would not be valid in the original equation with the square root. Therefore, we must always keep this condition in mind when manipulating and solving the equations. Recognizing the key features of the target equation, such as its quadratic form and the presence of a squared term, linear term, and constant term, is fundamental to solving this problem. This understanding will guide our approach as we analyze the given options and attempt to transform them into the desired form. By carefully considering the structure of the equation and the implications of each algebraic manipulation, we can efficiently determine the correct answer.

Analyzing the Options: A Step-by-Step Approach

Now, let's systematically analyze each of the given options to determine which one can be rewritten as x + 4 = x². We will apply algebraic manipulations, focusing on isolating the square root and then squaring both sides, to see if we can arrive at the target equation. This step-by-step approach will ensure that we consider each option thoroughly and avoid making hasty conclusions. The first option is: √(x) + 2 = x. To manipulate this equation, we first isolate the square root term by subtracting 2 from both sides: √(x) = x - 2. Now, we square both sides to eliminate the square root: (√(x))² = (x - 2)². This simplifies to x = x² - 4x + 4. Rearranging this equation to the standard quadratic form, we get: x² - 5x + 4 = 0. This equation is not equivalent to x² - x - 4 = 0, so the first option is not the correct answer. The second option is: √(x + 2) = x. To eliminate the square root, we square both sides: (√(x + 2))² = x². This simplifies to x + 2 = x². Rearranging this equation, we get: x² - x - 2 = 0. This equation is also not equivalent to x² - x - 4 = 0, so the second option is not the correct answer. The third option is: √(x + 4) = x. Squaring both sides, we get: (√(x + 4))² = x². This simplifies to x + 4 = x². Rearranging this equation, we get: x² - x - 4 = 0. This equation matches our target equation, so the third option is a potential answer. We still need to consider the constraint x > 0 and verify if the solution(s) to this equation satisfy the original equation. The fourth option is: √(x² + 16) = x. Squaring both sides, we get: (√(x² + 16))² = x². This simplifies to x² + 16 = x². Subtracting x² from both sides, we get 16 = 0, which is a contradiction. This means that there is no solution for this equation, and it cannot be rewritten as x + 4 = x². Through this systematic analysis, we have identified that the third option, √(x + 4) = x, can be rewritten as x + 4 = x². However, we still need to ensure that any solution we find for x satisfies the original equation and the condition x > 0. In the next section, we will delve into verifying the solution to ensure its validity.

Verifying the Solution: Ensuring Validity

Having identified √(x + 4) = x as the equation that can be rewritten as x + 4 = x², we now need to verify the solution(s) to ensure their validity. This step is crucial because squaring both sides of an equation can sometimes introduce extraneous solutions. We will solve the quadratic equation x² - x - 4 = 0 and then check if the solution(s) satisfy the original equation √(x + 4) = x and the condition x > 0. The quadratic equation x² - x - 4 = 0 does not factor easily, so we will use the quadratic formula to find the solutions. The quadratic formula is given by: x = [-b ± √(b² - 4ac)] / 2a. In our equation, a = 1, b = -1, and c = -4. Plugging these values into the quadratic formula, we get: x = [1 ± √((-1)² - 4(1)(-4))] / 2(1). Simplifying this, we have: x = [1 ± √(1 + 16)] / 2, which further simplifies to x = [1 ± √17] / 2. This gives us two possible solutions: x₁ = (1 + √17) / 2 and x₂ = (1 - √17) / 2. Since √17 is approximately 4.12, x₁ is approximately (1 + 4.12) / 2 = 2.56, which is greater than 0. However, x₂ is approximately (1 - 4.12) / 2 = -1.56, which is less than 0. Given the condition x > 0, we can discard x₂ as a valid solution. Now, we need to verify if x₁ = (1 + √17) / 2 satisfies the original equation √(x + 4) = x. Plugging x₁ into the original equation, we get: √([(1 + √17) / 2] + 4) = (1 + √17) / 2. Let's simplify the expression inside the square root: (1 + √17) / 2 + 4 = (1 + √17 + 8) / 2 = (9 + √17) / 2. So, we need to check if √[(9 + √17) / 2] = (1 + √17) / 2. Squaring both sides, we get: (9 + √17) / 2 = [(1 + √17) / 2]². Expanding the right side, we have: [(1 + √17) / 2]² = (1 + 2√17 + 17) / 4 = (18 + 2√17) / 4 = (9 + √17) / 2. Since both sides are equal, x₁ = (1 + √17) / 2 is a valid solution. Therefore, the equation √(x + 4) = x can indeed be rewritten as x + 4 = x², and the positive solution x₁ = (1 + √17) / 2 satisfies both equations. This verification process highlights the importance of checking solutions, especially when dealing with equations involving square roots. By systematically solving the quadratic equation and verifying the solution in the original equation, we can confidently conclude that the third option is the correct answer.

Conclusion: The Correct Equation

In conclusion, after a thorough analysis of the given options, we have successfully identified that the equation √(x + 4) = x can be rewritten as x + 4 = x², assuming x > 0. Our step-by-step approach involved manipulating each equation to isolate the square root, squaring both sides, and then comparing the resulting quadratic equation with our target equation. We found that squaring both sides of √(x + 4) = x directly leads to x + 4 = x², making it the correct answer. However, our analysis didn't stop there. We recognized the importance of verifying the solution(s) to the quadratic equation x² - x - 4 = 0 in the original equation. This is a critical step because squaring both sides can introduce extraneous solutions that do not satisfy the original equation. Using the quadratic formula, we found two potential solutions: x₁ = (1 + √17) / 2 and x₂ = (1 - √17) / 2. We discarded x₂ because it is negative and violates the condition x > 0. We then verified that x₁ indeed satisfies the original equation, confirming its validity. This process demonstrates the importance of a systematic and rigorous approach to solving algebraic problems. It's not enough to simply manipulate equations; we must also ensure that our solutions are valid and consistent with the given conditions. The techniques we've used in this guide, such as isolating square roots, squaring both sides, applying the quadratic formula, and verifying solutions, are fundamental tools in algebra. By mastering these techniques, you can confidently tackle a wide range of equation-solving problems. This problem also highlights the connection between different forms of equations. The ability to rewrite an equation in a different form while preserving its fundamental meaning is a powerful skill in mathematics. It allows us to approach problems from different angles and choose the most efficient method for finding a solution. In summary, the equation √(x + 4) = x is the correct answer, and our comprehensive analysis, including verification, provides a clear and convincing justification for this conclusion. Understanding the underlying principles and applying a systematic approach are key to success in algebra and beyond.

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