Solving The Equation Sqrt(u+6) = U A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving the equation √u+6=u, providing a step-by-step explanation to enhance understanding. We'll explore the necessary steps to arrive at the solution while highlighting potential pitfalls and ensuring the validity of the results. This exploration is crucial for students, educators, and anyone with an interest in mathematical problem-solving. Understanding how to manipulate square roots and quadratic equations is essential for success in algebra and beyond. We will also discuss the importance of checking solutions to avoid extraneous roots, which can arise when squaring both sides of an equation. By the end of this comprehensive guide, you will have a strong grasp of how to solve equations involving square roots and be able to apply these techniques to other similar problems.
The given equation, √u+6=u, involves a square root, making it a radical equation. To solve it, we need to isolate the square root and then eliminate it by squaring both sides. However, this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, it's crucial to check our solutions at the end of the process. This careful approach ensures that we only accept valid solutions. The equation itself represents a relationship between a variable, u, and its square root. Understanding this relationship is the first step towards finding the values of u that make the equation true. The equation's structure dictates the solution method: we must deal with the square root before we can isolate u. This process involves algebraic manipulation and careful attention to detail to ensure accuracy.
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Isolate the Square Root:
To begin, ensure the square root term is isolated on one side of the equation. In this case, √u+6 is already isolated on the left side, so no additional steps are needed for this part. Isolating the square root is a critical first step because it allows us to eliminate the radical by squaring both sides of the equation. This step simplifies the equation and brings us closer to finding a solution. Without isolating the square root first, squaring both sides would result in a more complex equation, making it harder to solve. The isolation step sets the stage for the next crucial action: eliminating the square root.
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Square Both Sides:
Squaring both sides of the equation eliminates the square root. This gives us:
(√u+6)² = u², which simplifies to u + 6 = u². Squaring both sides is a fundamental algebraic technique used to remove square roots from equations. However, this step is also where extraneous solutions can be introduced. When we square both sides, we are essentially creating a new equation that may have solutions that do not satisfy the original equation. This is because the squaring operation can make a negative value positive, potentially altering the solution set. Therefore, it is imperative to verify any solutions obtained after this step in the original equation to ensure their validity. The new equation, u + 6 = u², is a quadratic equation, which we can then solve using standard methods.
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Rearrange into a Quadratic Equation:
Rearrange the equation into the standard quadratic form, ax² + bx + c = 0. Subtracting u and 6 from both sides yields: 0 = u² - u - 6. This rearrangement is essential for applying methods such as factoring or the quadratic formula to solve the equation. The standard form provides a clear structure for identifying the coefficients a, b, and c, which are necessary for these solution techniques. By organizing the equation in this way, we can systematically approach the problem and avoid errors. The quadratic form also helps us recognize the potential for two solutions, as quadratic equations can have up to two distinct roots. This understanding guides our solution process, reminding us to look for both possible values of u.
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Solve the Quadratic Equation:
The quadratic equation u² - u - 6 = 0 can be solved by factoring. We look for two numbers that multiply to -6 and add to -1. These numbers are -3 and 2. Therefore, we can factor the equation as (u - 3)(u + 2) = 0. This factoring step is a key simplification that allows us to find the roots of the equation. Factoring breaks down the complex quadratic expression into a product of two simpler linear expressions. The solutions are found by setting each factor equal to zero. This technique is based on the principle that if the product of two numbers is zero, then at least one of the numbers must be zero. The factors (u - 3) and (u + 2) lead us to two potential solutions for u, which we must then verify in the original equation.
Setting each factor to zero gives us two possible solutions:
- u - 3 = 0 => u = 3
- u + 2 = 0 => u = -2
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Check for Extraneous Solutions:
It's crucial to check the solutions in the original equation to eliminate any extraneous solutions.
- For u = 3: √3 + 6 = √9 = 3. This solution is valid.
- For u = -2: √-2 + 6 = √4 = 2 ≠ -2. This solution is extraneous.
Checking for extraneous solutions is a critical final step in solving radical equations. Extraneous solutions can arise because squaring both sides of an equation can introduce solutions that do not satisfy the original equation. This is due to the fact that the squaring operation can eliminate the sign of a term, making a negative value appear positive. To check for extraneous solutions, each potential solution must be substituted back into the original equation. If the equation holds true, the solution is valid; otherwise, it is extraneous and must be discarded. In this case, u = -2 does not satisfy the original equation, so it is an extraneous solution. This verification process ensures that we only accept solutions that are mathematically sound.
Therefore, the only valid solution to the equation √u+6=u is u = 3. This process demonstrates the importance of not only solving the equation but also verifying the solutions to ensure accuracy. When solving equations involving square roots, it's essential to isolate the radical, square both sides, and, most importantly, check for extraneous solutions. This comprehensive approach ensures that we arrive at the correct answer and understand the underlying mathematical principles. The techniques used in solving this equation are applicable to a wide range of mathematical problems, making this a valuable skill to master. Remember that mathematics is not just about finding answers; it's about understanding the process and the reasons behind each step. By carefully following these steps and understanding the potential pitfalls, you can confidently solve radical equations and improve your overall mathematical proficiency. The practice of solving and verifying solutions enhances critical thinking and problem-solving skills, which are valuable in many areas of life.
