Solving The Equation √[2x+6]-4=0 A Step-by-Step Guide
In this article, we will delve into the process of solving the equation √[2x+6]-4=0 for the unknown variable x. This equation involves a radical, specifically a square root, making it a type of radical equation. Solving radical equations requires careful manipulation to isolate the variable while adhering to the rules of algebra. We'll break down each step in detail to ensure a clear understanding of the solution.
1. Isolate the Radical Term
The first crucial step in solving any radical equation is to isolate the radical term. This means getting the term containing the square root by itself on one side of the equation. In our case, the equation is:
√[2x+6]-4=0
To isolate the radical, we need to eliminate the '-4' from the left side. We can do this by adding 4 to both sides of the equation:
√[2x+6]-4+4=0+4
This simplifies to:
√[2x+6]=4
Now, the radical term √[2x+6] is isolated on the left side, setting us up for the next step.
2. Eliminate the Radical by Squaring Both Sides
With the radical isolated, the next step is to eliminate the radical. Since we have a square root, we can eliminate it by squaring both sides of the equation. Squaring a square root essentially undoes the radical operation.
(√[2x+6])^2 = 4^2
When we square the square root of (2x+6), we are left with just the expression inside the radical:
2x+6 = 4^2
And 4 squared is 16:
2x+6 = 16
Now we have a simple linear equation that we can easily solve for x.
3. Solve the Linear Equation
After eliminating the radical, we are left with a linear equation: 2x+6=16. To solve for x, we need to isolate x on one side of the equation. First, we subtract 6 from both sides:
2x+6-6 = 16-6
This simplifies to:
2x = 10
Now, to get x by itself, we divide both sides by 2:
2x/2 = 10/2
This gives us the solution:
x = 5
So, we have found a potential solution for x. However, with radical equations, it's crucial to perform one more step to ensure the validity of our solution.
4. Check the Solution
Checking the solution is a vital step in solving radical equations. This is because squaring both sides of an equation can sometimes introduce extraneous solutions – solutions that satisfy the transformed equation but not the original equation. To check our solution, we substitute x = 5 back into the original equation:
√[2x+6]-4=0
Substitute x = 5:
√[2(5)+6]-4=0
Simplify the expression inside the radical:
√[10+6]-4=0
√[16]-4=0
The square root of 16 is 4:
4-4=0
This simplifies to:
0=0
Since the equation holds true, our solution x = 5 is valid.
5. Conclusion and Answer
Therefore, the solution to the equation √[2x+6]-4=0 is x = 5. Among the given options:
A. -11 B. -1 C. 5 D. 6
The correct answer is C. 5.
In summary, solving radical equations involves isolating the radical, eliminating it by raising both sides to the appropriate power, solving the resulting equation, and most importantly, checking the solution to avoid extraneous results. This step-by-step process ensures an accurate solution to the given problem.
Let's further elaborate on each step involved in solving the radical equation √[2x+6]-4=0, providing a more in-depth understanding of the underlying principles.
Step 1: Isolate the Radical Term - The Foundation for Solving
The primary goal in solving any equation, especially those involving radicals, is to isolate the term containing the unknown variable. In the context of radical equations, this translates to isolating the radical expression. This isolation is crucial because it sets the stage for the next step: eliminating the radical itself.
Consider the given equation: √[2x+6]-4=0. The radical term here is √[2x+6]. This term is "attached" to the constant -4. Our immediate objective is to disentangle the radical term from this constant. To accomplish this, we employ the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain balance.
In this specific case, we observe that -4 is being subtracted from the radical term. The inverse operation of subtraction is addition. Therefore, we strategically add 4 to both sides of the equation. This operation effectively cancels out the -4 on the left side, leaving the radical term isolated. Mathematically, this can be represented as follows:
√[2x+6]-4+4=0+4
Upon simplification, the equation transforms into:
√[2x+6]=4
At this juncture, we have successfully isolated the radical term √[2x+6] on the left side of the equation. This isolation is a pivotal step, as it allows us to proceed with the elimination of the radical in the subsequent step.
The significance of isolating the radical term cannot be overstated. It streamlines the process of solving radical equations and paves the way for accurate solutions. Without this initial isolation, the subsequent steps would become significantly more complex and prone to errors. Therefore, mastering this step is essential for proficiency in solving radical equations.
Step 2: Eliminate the Radical by Squaring - Undoing the Root
Having successfully isolated the radical term √[2x+6] on one side of the equation, the next critical step is to eliminate the radical itself. In our equation, the radical is a square root. To counteract the square root, we employ its inverse operation: squaring.
The underlying principle here is that squaring a square root effectively "undoes" the radical. Mathematically, (√[a])^2 = a, where 'a' represents any non-negative expression. This principle forms the basis for eliminating square roots in equations.
To maintain the balance of the equation, we must apply the squaring operation to both sides. This ensures that the equality remains valid throughout the transformation. Thus, we square both sides of the equation √[2x+6]=4:
(√[2x+6])^2 = 4^2
On the left side, the square root and the square operation cancel each other out, leaving us with the expression inside the radical:
2x+6 = 4^2
On the right side, we evaluate 4 squared, which is 16:
2x+6 = 16
At this stage, the radical has been successfully eliminated, and we are left with a linear equation: 2x+6=16. This linear equation is significantly simpler to solve compared to the original radical equation. The elimination of the radical marks a crucial turning point in the solution process.
The importance of squaring both sides stems from the fundamental properties of radicals and exponents. Squaring a square root is analogous to raising a number to the power of 1/2 and then squaring the result, which effectively returns the original number. This concept is pivotal in manipulating radical expressions and solving equations involving them.
However, it is crucial to acknowledge that squaring both sides of an equation can sometimes introduce extraneous solutions. This phenomenon occurs because the squaring operation can transform an equation into a different equation with a broader solution set. Therefore, it is imperative to check the solutions obtained in the subsequent steps against the original equation to ensure their validity.
Step 3: Solve the Linear Equation - Isolating the Variable
With the radical successfully eliminated, we are now confronted with a linear equation: 2x+6=16. Solving a linear equation involves isolating the variable, in this case, x, on one side of the equation. This is achieved by performing a series of algebraic manipulations while maintaining the balance of the equation.
Our objective is to isolate x. The first step towards this goal is to eliminate the constant term, +6, from the left side of the equation. To do this, we employ the inverse operation of addition, which is subtraction. We subtract 6 from both sides of the equation:
2x+6-6 = 16-6
This operation cancels out the +6 on the left side, simplifying the equation to:
2x = 10
Now, we have a simpler equation with only the term 2x on the left side. To further isolate x, we need to eliminate the coefficient 2. The coefficient 2 is multiplying x, so we employ the inverse operation of multiplication, which is division. We divide both sides of the equation by 2:
2x/2 = 10/2
This operation cancels out the 2 on the left side, leaving us with x by itself:
x = 5
At this point, we have successfully isolated x and obtained a potential solution: x = 5. This solution represents the value of x that satisfies the linear equation 2x+6=16. However, it is crucial to remember that we derived this solution from a transformed equation, which was obtained by squaring both sides of the original radical equation. Therefore, we must verify this solution against the original equation to ensure its validity.
The process of solving a linear equation relies on the fundamental principles of algebraic manipulation. We strategically apply inverse operations to isolate the variable while preserving the equality. This step-by-step approach ensures that we arrive at the correct solution.
Step 4: Check the Solution - Ensuring Validity
As highlighted earlier, checking the solution is an indispensable step when solving radical equations. This stems from the fact that squaring both sides of an equation, while a valid algebraic manipulation, can sometimes introduce extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original radical equation. To safeguard against these spurious solutions, we must rigorously verify our potential solution against the original equation.
In our case, we obtained a potential solution of x = 5 for the equation √[2x+6]-4=0. To check this solution, we substitute x = 5 back into the original equation and evaluate both sides:
√[2(5)+6]-4=0
First, we simplify the expression inside the radical:
√[10+6]-4=0
√[16]-4=0
Next, we evaluate the square root of 16, which is 4:
4-4=0
Finally, we simplify the left side:
0=0
The result is a true statement: 0=0. This indicates that the potential solution x = 5 indeed satisfies the original equation. Therefore, x = 5 is a valid solution.
If, on the other hand, the substitution had resulted in a false statement, such as 1=0, it would have signified that the potential solution is extraneous and must be discarded. This underscores the critical role of the checking step in ensuring the accuracy of the solution.
The checking process involves a direct substitution of the potential solution into the original equation. This straightforward approach allows us to determine whether the value truly satisfies the equation or if it is an extraneous artifact of the algebraic manipulations. By diligently performing this check, we can confidently identify and discard any extraneous solutions, thereby arriving at the correct solution set.
Step 5: Conclusion and Answer - Finalizing the Solution
Having meticulously followed the steps of isolating the radical, eliminating the radical by squaring, solving the resulting linear equation, and, crucially, checking the solution, we arrive at the final conclusion: the solution to the equation √[2x+6]-4=0 is x = 5.
Among the provided options:
A. -11 B. -1 C. 5 D. 6
The correct answer is unequivocally C. 5.
This comprehensive process exemplifies the systematic approach required for solving radical equations. The key lies in the methodical application of algebraic principles, coupled with a keen awareness of the potential for extraneous solutions. Each step serves a specific purpose, contributing to the overall goal of isolating the variable and determining its value.
In summary, solving radical equations necessitates a multi-faceted strategy. It begins with the isolation of the radical term, followed by the elimination of the radical through the appropriate inverse operation (squaring in this case). The resulting equation, typically a linear equation, is then solved using standard algebraic techniques. However, the process is not complete until the potential solution is rigorously checked against the original equation. This crucial step ensures that we obtain only valid solutions, devoid of extraneous values.
The ability to solve radical equations is a valuable skill in mathematics, with applications spanning various fields. A thorough understanding of the underlying principles and a diligent application of the steps outlined herein will empower you to confidently tackle such equations and arrive at accurate solutions.
When solving radical equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:
- Forgetting to Check for Extraneous Solutions: This is the most common mistake. Always substitute your solution back into the original equation.
- Incorrectly Squaring: Make sure you square the entire side of the equation, not just individual terms.
- Algebraic Errors: Simple mistakes in adding, subtracting, multiplying, or dividing can lead to incorrect solutions. Double-check your work.
- Not Isolating the Radical: Always isolate the radical term before squaring. Squaring without isolating can make the equation more complex.
By avoiding these common errors, you can improve your accuracy and confidence in solving radical equations.
To solidify your understanding, try solving these practice problems:
- √(3x - 2) = 5
- √(x + 4) + 2 = 6
- √(5x + 1) - 3 = 1
Remember to follow the steps outlined in this article and check your solutions. Good luck!
By understanding each step thoroughly and practicing consistently, you can master the art of solving radical equations and improve your mathematical skills.
