How do you solve the equation √(2x + 1) = 3 for x, and how can you determine if the solution is extraneous?

Solving the Equation √(2x + 1) = 3

To solve the equation √(2x + 1) = 3, follow these steps:

1. Square both sides:
   Squaring both sides of the equation removes the square root:
   (√(2x + 1))² = 3²
   
   This simplifies to:
   2x + 1 = 9
2. Solve for x:
   Next, isolate x by first subtracting 1 from both sides:
   2x = 9 - 1
2x = 8
   
   Now, divide both sides by 2:
   x = 8 / 2
x = 4

Verifying the Solution

Now that we have a potential solution x = 4, we need to verify if it is valid:

Substituting x = 4 back into the original equation:

√(2(4) + 1) = √(8 + 1) = √9 = 3

This checks out, meaning x = 4 is indeed a solution.

Checking for Extraneous Solutions

An extraneous solution is one that does not satisfy the original equation due to the operations (like squaring) done during solving.

In this case, we found that substituting x = 4 back into the original equation gives us a true statement:

√(2x + 1) = 3

Since the equation holds true after checking it, we can conclude that x = 4 is not an extraneous solution.

Conclusion

The solved value is x = 4, and it is a valid solution to the original equation without any extraneous results.