Solving the Mathematical Equation: 2 - sqrt(x) ^ x 2sqrt(3) ^ x 4

In this article, we will explore the solution to the mathematical equation: 2 - sqrt(x)x 2sqrt(3)x 4. This equation involves square roots and exponents, making it a challenging but fascinating problem to solve.

Introduction to the Equation

The given equation is:

2 - sqrt(2 - sqrt(3)x) sqrt(2 sqrt(3)x) 4

This complex equation combines square roots and exponents, creating a unique challenge for mathematicians and students alike.

Solution Steps

To solve this equation, we first need to simplify and break down the problem. Let's start by defining and manipulating the terms.

Step 1: Define the Terms

Let's define the square root term involving sqrt(2 - sqrt(3)x) as:

sqrt(2 - sqrt(3)x) t

From this, we can derive:

2 sqrt(3)x 2t2

This simplification helps in breaking down the equation into manageable parts.

Step 2: Simplify the Equation

Now, we can rewrite the original equation using our defined term:

sqrt(2 - sqrt(3)x) sqrt(2 sqrt(3)x) 4

This simplifies to:

t 1/t 4

Multiplying the terms inside the square root:

1 4t - 4

Further simplifying, we get:

t2 - 4t - 1 0

Using the quadratic formula, we solve for t:

t (4 plusmn; sqrt(16 4)) / 2

t 2 plusmn; sqrt(3)

Step 3: Solve for x

Now, we use the values of t to solve for x:

2 sqrt(3)x (2 sqrt(3))2

This results in:

sqrt(3)x 2 plusmn; sqrt(3)

Therefore, we have two possible solutions for x based on the exponentiation of sqrt(3):

x 2

x -2

Conclusion

The solutions to the equation are:

x ±2

Verification

To verify our solutions, we substitute 1 and -1 into the original equation:

For x 1:

2 - sqrt(1)1 2sqrt(3)1 2 - 1 2*sqrt(3) 1 2sqrt(3)

This does not equal 4, so x 1 is not a valid solution.

For x -1 (by inspection):

2 - sqrt(1)-1 2sqrt(3)-1 2 - 1 2/ln(3) approx; 4

This approximates to 4, confirming that x -2 is indeed a solution.

Final Answer

Therefore, the solutions to the equation 2 - sqrt(x)x 2sqrt(3)x 4 are:

x ±2