Solving Radical Equation √x-6 + 3 = 10 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. When equations involve radicals, such as square roots, the process requires careful attention to isolate the variable. This article provides a comprehensive guide on how to solve the equation √x-6 + 3 = 10, demonstrating each step with clarity and precision.
Understanding Radical Equations
Radical equations are equations that contain a variable within a radical expression, most commonly a square root. The key to solving these equations lies in isolating the radical term and then eliminating the radical by raising both sides of the equation to the appropriate power. In the given equation, √x-6 + 3 = 10, the radical term is √x-6. Our goal is to isolate this term before we can proceed to solve for x.
Step 1: Isolate the Radical
Isolating the radical term is the first and crucial step. To do this, we need to eliminate any terms that are outside the radical on the same side of the equation. In our equation, we have +3 outside the radical. To eliminate this, we subtract 3 from both sides of the equation:
√x-6 + 3 - 3 = 10 - 3
This simplifies to:
√x-6 = 7
Now, the radical term, √x-6, is isolated on one side of the equation. This sets us up for the next step, which involves eliminating the radical.
Step 2: Eliminate the Radical
To eliminate the square root, we square both sides of the equation. This is because the square of a square root cancels out the radical, leaving us with the expression inside the radical:
(√x-6)² = 7²
Squaring both sides gives us:
x - 6 = 49
Now, we have a simple linear equation that is much easier to solve.
Step 3: Solve for x
To solve for x, we need to isolate x on one side of the equation. In this case, we have x - 6 = 49. To isolate x, we add 6 to both sides of the equation:
x - 6 + 6 = 49 + 6
This simplifies to:
x = 55
So, we have found a potential solution for x. However, with radical equations, it is essential to check our solution to ensure it is valid.
Step 4: Check the Solution
Checking the solution is a critical step in solving radical equations. This is because squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. To check our solution, we substitute x = 55 back into the original equation:
√55-6 + 3 = 10
Simplify the expression inside the square root:
√49 + 3 = 10
Evaluate the square root:
7 + 3 = 10
Simplify:
10 = 10
Since the equation holds true, our solution x = 55 is valid.
Common Mistakes and How to Avoid Them
Solving radical equations can be tricky, and there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and solve equations more accurately.
Mistake 1: Forgetting to Isolate the Radical
The most common mistake 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 isolate the radical term first.
For example, if we had squared both sides of the original equation (√x-6 + 3 = 10) without isolating the radical, we would have:
(√x-6 + 3)² = 10²
This expands to:
(x - 6) + 6√x-6 + 9 = 100
This equation is significantly more complicated to solve than the one we obtained by isolating the radical first.
Mistake 2: Squaring Terms Individually
Another common mistake is squaring terms individually when there is a binomial. Remember that (a + b)² is not equal to a² + b². Instead, it is equal to a² + 2ab + b². When squaring both sides of the equation, make sure to expand the binomial correctly.
Mistake 3: Forgetting to Check for Extraneous Solutions
As mentioned earlier, squaring both sides of an equation can introduce extraneous solutions. Therefore, it is crucial to check your solution in the original equation. If the solution does not satisfy the original equation, it is an extraneous solution and should be discarded.
Mistake 4: Incorrectly Simplifying Radicals
Simplifying radicals incorrectly can lead to errors in the solution. Make sure you understand the properties of radicals and how to simplify them correctly. For instance, √a² = |a|, not just a. This is especially important when dealing with variables.
Tips for Solving Radical Equations
To improve your skills in solving radical equations, consider the following tips:
- Always isolate the radical first: This is the most important step in solving radical equations. It simplifies the process and reduces the chances of making mistakes.
- Square both sides carefully: When squaring both sides of the equation, make sure to expand any binomials correctly.
- Check your solutions: Always check your solutions in the original equation to eliminate extraneous solutions.
- Simplify radicals whenever possible: Simplifying radicals can make the equation easier to work with.
- Practice regularly: The more you practice, the more comfortable you will become with solving radical equations.
Advanced Radical Equations
While the equation √x-6 + 3 = 10 is a relatively simple example, radical equations can become more complex. For instance, some equations may involve multiple radicals or radicals on both sides of the equation. The same principles apply to solving these equations, but the steps may need to be repeated or combined.
Equations with Multiple Radicals
Consider an equation with two radicals, such as √(x + 1) + √(x - 2) = 3. To solve this equation, you would first isolate one of the radicals. Then, square both sides of the equation. This will eliminate one of the radicals, but you will still have one radical left. Isolate the remaining radical and square both sides again. Finally, solve the resulting equation and check your solutions.
Equations with Radicals on Both Sides
For equations with radicals on both sides, such as √(2x + 1) = √(x + 4), you can square both sides immediately to eliminate both radicals. Then, solve the resulting equation and check your solutions.
Fractional Exponents
Radicals can also be expressed using fractional exponents. For example, √x is the same as x^(1/2). Equations with fractional exponents can be solved using the same principles as equations with radicals. Raise both sides of the equation to the reciprocal of the fractional exponent to eliminate the exponent. For example, to solve x^(1/2) = 5, raise both sides to the power of 2:
(x^(1/2))² = 5²
This simplifies to:
x = 25
Conclusion
Solving radical equations requires a systematic approach. By isolating the radical, eliminating the radical, solving for the variable, and checking the solution, you can confidently tackle these types of equations. Remember to be mindful of common mistakes and practice regularly to enhance your skills. Whether you are dealing with simple or complex radical equations, the principles outlined in this guide will help you find accurate solutions. Mastering radical equations is a valuable skill in mathematics, opening doors to more advanced topics and problem-solving techniques.
