Solving Mathematical Puzzles: Techniques and Approaches

Mathematical puzzles often present intriguing challenges that require a mix of creativity and analytical skills. In this article, we will explore the solution to a specific math puzzle and discuss the techniques involved in solving such problems. Let's break down the puzzle and examine the step-by-step process to find the solution.

The Puzzle

The puzzle in question involves the following equation:

2^x 1 - 3 2^(2x) - 5

Step-by-Step Solution

Here's a detailed walkthrough of the solution:

Step 1: Make Both Equations Equal

2^x 1 - 3 2^(2x) - 5

2^x 1 - 3 3 2^(2x) - 5 3

2^x 1 2^(2x) - 2

Step 2: Represent -2 as a Power of Two

2^x 1 2^(2x) - 2^1

Step 3: Take Away All of the Bases

x 1 2x - 1

Step 4: Subtract 2x from Both Sides

x 1 - 2x 2x - 1 - 2x

-x 1 -1

Step 5: Subtract 1 from Both Sides

-x 1 - 1 -1 - 1

-x -2

Step 6: Multiply Both Sides by -1

(-1)(-x) (-1)(-2)

x 2

Thus, the solution to the equation is x 2.

Alternative Approaches

Another approach to solving this puzzle, as described in the second part, involves using a substitution:

y 2^x

The transformed equation is:

8 y^2 - 2y

Factoring the quadratic equation:

8 y(y - 2)

Setting each factor to zero:

y 1 ± 3

Given the real domain, we can rule out the negative solution, resulting in:

y 4

Therefore:

2^x 4

Which gives us:

x 2

Dealing with Quadratics and Complex Solutions

In the context of quadratic equations, it's crucial to be prepared for complex solutions. A more complex approach to solving the given quadratic equation using MATLAB involves:

roots([1, -2, -8])

Which outputs:

ans 4 - 2i

While handling logs, the equation:

log(4 - 2i) / log(2)

Yields:

2.0000 0.0000i

And the complex solution:

1 4.5324i

For principal solutions, setting n m 0 gives:

dfrac {2 2 pi i n mid n in Z} {1 2 pi i m mid m in Z}, and the other

dfrac {1 4.5324i 2 pi i n mid n in Z} {1 2 pi i m mid m in Z}

Conclusion

In conclusion, mathematical puzzles like these require a systematic approach and an understanding of both algebraic and logarithmic principles. Whether through manual methods or computational tools, the key is to break down the problem into manageable steps and consider all possible solutions, including complex ones.

References

1. MATLAB Documentation for 'roots' function.

2. Complex solutions in logarithmic equations.