Solving For Q: A Step-by-Step Guide

Let's dive into solving this equation step-by-step. The equation we're tackling is √(4q - 2) = √(3q + 8). Our mission is to isolate 'q' and find its value. This involves a bit of algebraic maneuvering, but don't worry, we'll break it down into easy-to-follow steps. Understanding the basics of algebraic equations is crucial before we start. Remember, what we do on one side of the equation, we must also do on the other side to maintain balance. This principle is the bedrock of solving for any variable. Moreover, being comfortable with square roots and how they interact with squares will be extremely helpful. In this particular problem, we will utilize the property that squaring a square root cancels out, leaving us with the expression inside the square root. Before we jump into the detailed solution, ensure you have a pen and paper ready to follow along and maybe try solving it yourself as we proceed. Understanding each step conceptually is more important than just memorizing the process. As you practice more problems like this, you'll develop a better intuition for algebraic manipulations, making it easier to solve more complex equations in the future.

Step 1: Squaring Both Sides

The first crucial step in solving the equation √(4q - 2) = √(3q + 8) is to eliminate the square roots. To do this, we square both sides of the equation. Squaring both sides maintains the equality and removes the square root, simplifying the equation significantly. When we square the left side, (√(4q - 2))^2, the square root and the square cancel each other out, leaving us with 4q - 2. Similarly, when we square the right side, (√(3q + 8))^2, the square root and the square cancel each other out, leaving us with 3q + 8. Therefore, after squaring both sides, the equation becomes:

4q - 2 = 3q + 8

This new equation is much easier to work with because it no longer contains square roots. Our next goal is to isolate the variable 'q' on one side of the equation. This involves moving the terms containing 'q' to one side and the constant terms to the other side. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to keep the equation balanced. This ensures that the value of 'q' we find is the correct solution to the original equation. Make sure to double-check your work as you proceed, as a simple mistake in the early steps can lead to an incorrect final answer. With the square roots now out of the way, we can confidently move forward to isolate and solve for 'q'.

Step 2: Isolating q

Now that we have the equation 4q - 2 = 3q + 8, our next aim is to isolate 'q' on one side of the equation. To do this, we can subtract 3q from both sides. This will move the 'q' term from the right side to the left side. Subtracting 3q from both sides of the equation gives us:

4q - 3q - 2 = 3q - 3q + 8

Simplifying this, we get:

q - 2 = 8

Now, to completely isolate 'q', we need to get rid of the -2 on the left side. We can do this by adding 2 to both sides of the equation. Adding 2 to both sides gives us:

q - 2 + 2 = 8 + 2

Simplifying this, we get:

q = 10

So, we've found a potential solution for 'q', which is q = 10. However, it's crucial to verify this solution by plugging it back into the original equation. This ensures that our solution is valid and doesn't introduce any extraneous roots due to the squaring operation we performed earlier. Remember, always double-check your solutions, especially when dealing with square roots or other operations that can sometimes lead to false solutions. Accuracy and verification are key to mastering algebraic equations.

Step 3: Verification

After finding a potential solution, it's essential to verify it. Substitute q = 10 back into the original equation √(4q - 2) = √(3q + 8) to check if it holds true. Replacing 'q' with 10, we get:

√(4(10) - 2) = √(3(10) + 8)

Simplify the expressions inside the square roots:

√(40 - 2) = √(30 + 8)

√(38) = √(38)

Since both sides of the equation are equal, our solution q = 10 is valid. Verification is a critical step in solving equations, especially those involving square roots or rational expressions. This process helps to ensure that the solution obtained does not introduce any extraneous roots, which are solutions that satisfy the transformed equation but not the original one. In this case, substituting q = 10 into the original equation yields a true statement, confirming that it is indeed a valid solution. By verifying our answer, we can be confident that we have solved the equation correctly. Always remember to take the time to verify your solutions to avoid errors and ensure accuracy in your mathematical work.

Conclusion

Therefore, the value of q in the equation √(4q - 2) = √(3q + 8) is 10. By following the steps of squaring both sides, isolating 'q', and verifying the solution, we've successfully solved the equation. Always remember to verify your solutions, especially when dealing with square roots. Understanding the importance of each step and practicing consistently will help you master solving algebraic equations. Keep practicing and exploring more problems to build your confidence and skills in algebra. Remember, every problem you solve is a step forward in your mathematical journey. Don't be discouraged by difficult problems; instead, see them as opportunities to learn and grow. With persistence and a solid understanding of the fundamental principles, you can tackle any algebraic challenge that comes your way. Happy solving!

For more information on solving algebraic equations, you can visit Khan Academy's Algebra Section.