Solving The Radical Equation √(2x - 7) - 3 = 4 A Step By Step Guide
This article provides a detailed walkthrough on how to solve the radical equation √(2x - 7) - 3 = 4. We will explore the step-by-step process, explain the underlying concepts, and discuss how to verify the solutions. This guide aims to provide a comprehensive understanding of solving radical equations and will be helpful for students and anyone interested in mathematics.
Understanding Radical Equations
Radical equations are algebraic equations where the variable is under a radical, most commonly a square root. Solving these equations involves isolating the radical term and then eliminating the radical by raising both sides of the equation to the appropriate power. However, it's crucial to check the solutions obtained because this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Extraneous solutions arise because the squaring operation can turn negative values into positive values, potentially changing the nature of the equation. Therefore, verification is an essential part of the solution process.
Step-by-Step Solution
1. Isolate the Radical
The first step in solving the equation √(2x - 7) - 3 = 4 is to isolate the radical term. This involves moving the constant term (-3) to the other side of the equation. To do this, we add 3 to both sides:
√(2x - 7) - 3 + 3 = 4 + 3
This simplifies to:
√(2x - 7) = 7
Now, the square root term is isolated on one side of the equation.
2. Eliminate the Radical
To eliminate the square root, we square both sides of the equation. Squaring both sides will undo the square root, allowing us to solve for x:
(√(2x - 7))² = 7²
This simplifies to:
2x - 7 = 49
3. Solve for x
Now that the radical is eliminated, we have a simple linear equation. To solve for x, we first add 7 to both sides:
2x - 7 + 7 = 49 + 7
This simplifies to:
2x = 56
Next, we divide both sides by 2:
2x / 2 = 56 / 2
This gives us:
x = 28
4. Check the Solution
It is essential to check the solution in the original equation to ensure it is not extraneous. Substitute x = 28 back into the original equation:
√(2(28) - 7) - 3 = 4
Simplify inside the square root:
√(56 - 7) - 3 = 4
√(49) - 3 = 4
7 - 3 = 4
4 = 4
Since the equation holds true, x = 28 is a valid solution.
Common Mistakes and How to Avoid Them
1. Forgetting to Isolate the Radical
One of the most common mistakes is squaring both sides of the equation before isolating the radical. This can lead to a more complex equation that is difficult to solve. Always make sure to isolate the radical term first.
2. Incorrectly Squaring Both Sides
When squaring both sides of an equation, it is crucial to correctly apply the distributive property if there are multiple terms. For example, if the equation were (√(2x - 7) - 3)² = 4², you would need to expand the left side correctly, which is different from just squaring each term individually.
3. Not Checking for Extraneous Solutions
As mentioned earlier, not checking for extraneous solutions is a significant error. Always substitute the solution back into the original equation to verify its validity.
4. Arithmetic Errors
Simple arithmetic errors can occur during the solving process. Double-check each step to ensure accuracy, especially when dealing with negative numbers or fractions.
Advanced Tips and Tricks
1. Dealing with Multiple Radicals
If an equation contains multiple radicals, isolate one radical at a time and eliminate it by raising both sides to the appropriate power. Repeat this process until all radicals are eliminated.
2. Substitution
In some cases, substitution can simplify the equation. For example, if you have a complex radical expression, you might substitute a single variable for that expression to make the equation easier to solve.
3. Recognizing Patterns
Sometimes, radical equations can be transformed into quadratic equations. Recognizing these patterns can help you apply appropriate methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula.
Real-World Applications
Radical equations are not just abstract mathematical concepts; they have real-world applications in various fields, including:
1. Physics
In physics, radical equations are used to calculate the speed of an object, the period of a pendulum, and other physical quantities. For example, the formula for the period T of a simple pendulum is T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity.
2. Engineering
Engineers use radical equations in structural design, fluid dynamics, and other areas. For instance, the stress on a structural component can be related to the square root of the applied force.
3. Computer Graphics
Radical equations are used in computer graphics to calculate distances and transformations, which are essential for rendering 3D images and animations.
4. Finance
In finance, radical equations can be used to calculate the rate of return on an investment or the time it takes for an investment to reach a certain value.
Practice Problems
To reinforce your understanding, here are a few practice problems:
- √(3x + 4) = 5
- √(x - 2) + 3 = 7
- √(4x + 1) - 2 = 0
- √(5x - 6) = √(2x + 3)
- √(x + 1) + √(x - 1) = 2
Work through these problems, applying the steps outlined in this guide. Remember to check your solutions to avoid extraneous roots.
Conclusion
Solving radical equations requires a systematic approach. By isolating the radical, eliminating it through squaring, solving the resulting equation, and checking for extraneous solutions, you can confidently solve a wide range of radical equations. This article has provided a thorough guide, including step-by-step solutions, common mistakes to avoid, advanced tips, real-world applications, and practice problems. With practice, solving radical equations can become a manageable and even enjoyable mathematical exercise. Remember, the key is to be methodical and meticulous in each step.
Therefore, the correct answer is D. 28
