Solving The Equation √[4](x² - 7) = 2 A Step By Step Guide
Radical equations, which involve variables within radicals (square roots, cube roots, fourth roots, etc.), might seem intimidating at first. However, with a systematic approach, these equations can be solved effectively. This article delves into the process of solving the specific equation √[4](x² - 7) = 2, providing a step-by-step guide and exploring the underlying mathematical concepts. We will also discuss how to verify the solutions and touch upon common pitfalls to avoid. Mastering the art of solving radical equations is crucial for various fields, including algebra, calculus, and physics.
Understanding Radical Equations
Before diving into the solution, it's crucial to grasp the fundamental concepts of radical equations. A radical equation is an equation where the variable appears inside a radical expression. The most common type is the square root, denoted by √, but radicals can also include cube roots (∛), fourth roots (√[4]), and so on. The number indicating the type of root is called the index. For example, in √[4](x² - 7), the index is 4, signifying a fourth root. The expression inside the radical, in this case, x² - 7, is known as the radicand.
The key to solving radical equations lies in isolating the radical term and then eliminating the radical by raising both sides of the equation to a power equal to the index. This process is based on the principle that if a = b, then aⁿ = bⁿ for any positive integer n. However, it is imperative to check the solutions obtained, as this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This typically occurs when dealing with even-indexed radicals due to the nature of even powers.
Step-by-Step Solution of √[4](x² - 7) = 2
Let's embark on a detailed solution of the equation √[4](x² - 7) = 2:
1. Isolate the Radical Term
The first step in solving any radical equation is to isolate the radical term on one side of the equation. In our case, the radical term, √[4](x² - 7), is already isolated on the left side. This simplifies our initial setup and allows us to proceed directly to the next step.
2. Eliminate the Radical
To eliminate the fourth root, we need to raise both sides of the equation to the power of 4, which is the index of the radical. This step effectively "undoes" the radical operation. We have:
(√[4](x² - 7))⁴ = 2⁴
This simplifies to:
x² - 7 = 16
3. Solve the Resulting Equation
Now, we have a simpler equation to solve, which is a quadratic equation. To solve for x, we first need to isolate the x² term. Add 7 to both sides of the equation:
x² - 7 + 7 = 16 + 7
x² = 23
Next, to solve for x, we take the square root of both sides. Remember to consider both the positive and negative square roots, as both values will satisfy the equation:
x = ±√23
So, we have two potential solutions: x = √23 and x = -√23.
4. Check for Extraneous Solutions
This is a crucial step that cannot be skipped, especially when dealing with even-indexed radicals. We need to substitute each potential solution back into the original equation, √[4](x² - 7) = 2, to ensure it satisfies the equation. Extraneous solutions can arise because raising both sides of an equation to an even power can introduce solutions that do not work in the original equation.
Checking x = √23
Substitute x = √23 into the original equation:
√[4]((√23)² - 7) = 2
√[4](23 - 7) = 2
√4 = 2
2 = 2
This solution is valid.
Checking x = -√23
Substitute x = -√23 into the original equation:
√[4]((-√23)² - 7) = 2
√[4](23 - 7) = 2
√4 = 2
2 = 2
This solution is also valid.
The Solutions
Both potential solutions, x = √23 and x = -√23, satisfy the original equation. Therefore, the solutions to the equation √[4](x² - 7) = 2 are x = √23 and x = -√23.
Common Mistakes to Avoid
Solving radical equations can be tricky, and there are several common mistakes that students often make. Awareness of these pitfalls can help you avoid them and ensure accurate solutions.
Forgetting to Check for Extraneous Solutions
As emphasized earlier, this is the most critical step, especially when dealing with even-indexed radicals. Failing to check solutions can lead to including extraneous solutions in your answer, which are not valid solutions to the original equation. Always substitute your potential solutions back into the original equation to verify their validity.
Incorrectly Applying the Power
When eliminating the radical, it's essential to raise the entire side of the equation to the power, not just individual terms. For example, if you had an equation like √(x + 2) = x, you would square the entire left side, (√(x + 2))², and the entire right side, x², not just the x term inside the radical. This ensures that you are maintaining the equality of the equation.
Making Algebraic Errors
Solving radical equations often involves multiple algebraic steps, such as simplifying expressions, combining like terms, and factoring. Making errors in these steps can lead to incorrect solutions. Carefully review each step and double-check your work to minimize the chance of algebraic mistakes.
Not Isolating the Radical First
Isolating the radical term before raising both sides to a power is crucial. If you don't isolate the radical, you might end up with a more complex equation that is difficult to solve. For instance, if you have an equation like √(x + 1) + 2 = 5, you should first subtract 2 from both sides to isolate the radical: √(x + 1) = 3. Then, you can square both sides.
Additional Tips for Solving Radical Equations
To further enhance your problem-solving skills, consider these additional tips:
- Simplify Radicals: Before attempting to solve the equation, simplify any radicals if possible. This can make the equation easier to work with.
- Use Substitution: In some cases, substitution can simplify the equation. For example, if you have an equation with a complex radical expression, you might substitute a variable for that expression to make the equation more manageable.
- Practice Regularly: The best way to improve your skills is to practice solving a variety of radical equations. Work through examples, try different problems, and review your work to identify any areas where you need improvement.
Conclusion
Solving radical equations requires a systematic approach, including isolating the radical, eliminating the radical by raising both sides to the appropriate power, solving the resulting equation, and, most importantly, checking for extraneous solutions. By following these steps and avoiding common mistakes, you can confidently solve radical equations and enhance your mathematical problem-solving skills. The equation √[4](x² - 7) = 2 serves as a great example to illustrate this process, with the solutions being x = √23 and x = -√23. Remember to always verify your solutions to ensure accuracy and deepen your understanding of the underlying mathematical principles.
By understanding the principles of solving radical equations and applying the steps outlined in this article, you can confidently tackle a wide range of problems. Practice consistently and remember to check your solutions to avoid extraneous answers. Solving radical equations is a valuable skill that enhances your mathematical proficiency and prepares you for more advanced concepts in algebra and beyond.