Solving Radical Equations Step By Step Guide

In this article, we will walk through the step-by-step solution of the given equation: $\sqrt{w^2-20}-2=2$. We will explore the algebraic techniques required to isolate the variable w and determine the correct solutions. Additionally, we will analyze each potential solution to ensure it satisfies the original equation, thereby avoiding any extraneous roots. Our goal is to provide a comprehensive explanation that enhances understanding and enables you to confidently solve similar equations. We will also discuss common pitfalls in solving such equations and strategies to avoid them. Through detailed steps and explanations, we aim to make this process as clear and straightforward as possible.

To solve the equation $\sqrt{w^2-20}-2=2$, we'll follow these steps:

Step 1: Isolate the Square Root

First, we need to isolate the square root term. To do this, we add 2 to both sides of the equation:

$$\sqrt{w^2-20}-2+2=2+2$$

$$\sqrt{w^2-20}=4$$

This step simplifies the equation by getting the square root term by itself on one side. Isolating the square root is a crucial first step because it allows us to eliminate the square root by squaring both sides, which we will do in the next step. By focusing on isolating the square root, we make the subsequent steps more manageable and reduce the chances of making errors.

Step 2: Square Both Sides

To eliminate the square root, we square both sides of the equation:

$$(\sqrt{w^2-20})^2=4^2$$

$$w^2-20=16$$

Squaring both sides is a standard technique for solving equations involving square roots. By squaring, we transform the equation into a simpler form that we can solve using algebraic methods. It's important to remember that squaring both sides can sometimes introduce extraneous solutions, so we will need to check our final answers in the original equation to ensure they are valid.

Step 3: Isolate the Variable Term

Next, we isolate the ${ w^2 }$ term by adding 20 to both sides:

$$w^2-20+20=16+20$$

$$w^2=36$$

This step brings us closer to solving for w by isolating the squared term. Adding 20 to both sides maintains the equation's balance and simplifies it further. The goal here is to get ${ w^2 }$ by itself so that we can easily take the square root in the next step. Isolating the variable term is a common strategy in algebra to simplify equations and make them easier to solve.

Step 4: Solve for w

To solve for w, we take the square root of both sides:

$$\sqrt{w^2}=\sqrt{36}$$

$$w=\pm 6$$

This step gives us two possible solutions: w = 6 and w = -6. Taking the square root of both sides is the final step in solving for w. It’s crucial to remember that when we take the square root of a number, we consider both the positive and negative roots, which is why we have ${\pm 6}$. However, we still need to check these solutions in the original equation to make sure they are valid.

Step 5: Check the Solutions

Now, we need to check both solutions in the original equation to ensure they are valid.

Check w = 6

Substitute w = 6 into the original equation:

$$\sqrt{6^2-20}-2=2$$

$$\sqrt{36-20}-2=2$$

$$\sqrt{16}-2=2$$

$$4-2=2$$

$$2=2$$

Since the equation holds true, w = 6 is a valid solution.

Check w = -6

Substitute w = -6 into the original equation:

$$\sqrt{(-6)^2-20}-2=2$$

$$\sqrt{36-20}-2=2$$

$$\sqrt{16}-2=2$$

$$4-2=2$$

$$2=2$$

Since the equation also holds true, w = -6 is also a valid solution.

Conclusion

Both w = 6 and w = -6 satisfy the original equation. Checking solutions is a critical step in solving equations with square roots to avoid extraneous solutions, which are solutions that arise from the solving process but do not satisfy the original equation. In this case, both solutions are valid because they make the original equation true.

The solutions to the equation $\sqrt{w^2-20}-2=2$ are w = -6 and w = 6.

The correct answers are:

• A. -6
• D. 6

When solving equations involving square roots, there are several common mistakes that students often make. Being aware of these mistakes and knowing how to avoid them can significantly improve your accuracy and confidence in solving these types of problems. In this section, we will discuss some of the most frequent errors and provide strategies to help you avoid them.

1. Forgetting to Check for Extraneous Solutions

The Mistake

One of the most common errors is forgetting to check the solutions in the original equation. Squaring both sides of an equation can introduce extraneous solutions, which are solutions that do not satisfy the original equation. Failing to check can lead to including incorrect solutions in your final answer.

How to Avoid It

Always substitute the solutions you obtain back into the original equation to verify their validity. If a solution does not satisfy the original equation, it is an extraneous solution and should be discarded. This step is crucial for ensuring that your final answer is accurate.

2. Incorrectly Squaring Both Sides

The Mistake

Another frequent mistake is squaring both sides of the equation incorrectly. For instance, if the equation has multiple terms on one side, you need to ensure you are squaring the entire side, not just individual terms. Incorrectly squaring can lead to a completely different equation and, consequently, incorrect solutions.

How to Avoid It

When squaring both sides, make sure to square the entire expression. If there are multiple terms, use the correct algebraic identity (e.g., ${(a + b)^2 = a^2 + 2ab + b^2}$) or use the distributive property (FOIL method) to expand correctly. Taking the time to double-check your squaring process can prevent this error.

3. Neglecting the Negative Root

The Mistake

When taking the square root of both sides of an equation, it is essential to consider both the positive and negative roots. Neglecting the negative root can cause you to miss one of the solutions, leading to an incomplete answer.

How to Avoid It

Always remember to include both the positive and negative square roots when solving for a variable. For example, if ${ w^2 = 36 }$, then ${ w = \pm 6 }$. Considering both roots ensures you find all possible solutions.

4. Misunderstanding the Order of Operations

The Mistake

Sometimes, mistakes occur due to a misunderstanding of the order of operations. Failing to isolate the square root term before squaring both sides can complicate the problem and lead to errors. Misapplying the order of operations can result in an equation that is much harder to solve.

How to Avoid It

Always follow the correct order of operations (PEMDAS/BODMAS). Isolate the square root term first before squaring both sides. This simplifies the equation and makes the subsequent steps easier to manage. Carefully review each step to ensure you are applying the correct operations in the correct order.

5. Arithmetic Errors

The Mistake

Simple arithmetic errors can also lead to incorrect solutions. These can occur during any step of the process, such as adding or subtracting terms, squaring numbers, or simplifying expressions. Arithmetic errors, though seemingly minor, can significantly impact the final result.

How to Avoid It

Take your time and double-check your calculations at each step. Use a calculator if necessary to verify your arithmetic. Writing down each step clearly and methodically can also help you spot and correct any errors more easily.

To effectively solve equations involving square roots and avoid common mistakes, consider the following tips:

• Practice Regularly: The more you practice, the more comfortable and proficient you will become at solving these types of equations.
• Show Your Work: Writing down each step helps you stay organized and makes it easier to identify and correct errors.
• Double-Check Your Work: Always review your solution process and calculations to ensure accuracy.
• Understand the Concepts: Make sure you understand the underlying algebraic principles and techniques. This will enable you to approach problems with confidence.
• Seek Help When Needed: If you are struggling, don't hesitate to ask for help from a teacher, tutor, or classmate.

By being aware of these common mistakes and following the tips for success, you can significantly improve your ability to solve equations involving square roots accurately and efficiently.