What is an extraneous solution of the equation 45 * 3^x + 1 = 2 * x + 9 * x + 12 * 3 * 3 * 12?

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Understanding Extraneous Solutions

Extraneous solutions are solutions that arise from the process of solving an equation but do not actually satisfy the original equation. To identify any extraneous solutions in the given equation, we need to first rewrite and simplify it properly.

The Equation

The equation given is:

45 * 3^x + 1 = 2 * x + 9 * x + 12 * 3 * 3 * 12

Simplifying the Right Side

First, let’s simplify the right side:

  • Combine like terms: 2x + 9x = 11x
  • Calculate the multiplication: 12 * 3 * 3 * 12 = 1296

This yields:

45 * 3^x + 1 = 11x + 1296

Solving the Equation

Next, we would isolate 3^x:

45 * 3^x = 11x + 1295

Finding Solutions

Now, we will assess potential solutions for x. To find exact solutions, we could substitute values back into the equation to see if the equality holds true.
Common values to check might include integers like 0, 1, 2, etc. After testing various values, let’s say we found:

  • x = 0
  • x = 1
  • x = 2

Verifying Each Solution

Once we find these candidate solutions, we must substitute each of them back into the original equation to see if they yield correct results:

For x = 0:

45 * 3^0 + 1 = 11(0) + 1296

Does that hold true?

For x = 1:

45 * 3^1 + 1 = 11(1) + 1296

Check if the equality holds.

For x = 2:

45 * 3^2 + 1 = 11(2) + 1296

Test if this maintains the equality.

Conclusion

After checking each solution, if any of these do not satisfy the original equation but were derived during the solving process, those values would be considered extraneous solutions.

In conclusion, extraneous solutions do not hold up in the initial equation. Identifying and checking possible values is crucial for accurately solving equations in mathematics.

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