Solving the Equation: sqrt{-G1^2} Gsqrt{2}

In this article, we delve into the intricate process of solving the equation sqrt{-G1^2} Gsqrt{2}. This equation involves complex numbers and requires a nuanced approach to ensure accurate and correct solutions. We will explore the algebraic manipulation and reasoning needed to arrive at a valid solution.

1. Introduction to the Equation

The given equation is sqrt{-G1^2} Gsqrt{2}. At first glance, this equation might seem intimidating due to the presence of the negative sign and square root. However, by breaking it down step-by-step, we can solve it effectively.

2. Square Both Sides

Let's start by squaring both sides of the equation to eliminate the square root on the left-hand side:

sqrt{-G1^2} Gsqrt{2}

sqrt{-G1^2}^2 (Gsqrt{2})^2

-G1^2 G^2 * 2

-G1^2 2G^2

Rearrange the equation to form a standard quadratic equation:

2G^2 G1^2 0

3. Solve the Quadratic Equation

We can solve the quadratic equation 2G^2 G1^2 0 using the quadratic formula:

G dfrac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 2, b 0, and c 1. Substituting these values into the formula, we get:

G dfrac{0 pm sqrt{0^2 - 4 * 2 * 1}}{2 * 2}

Since the discriminant (0^2 - 4 * 2 * 1) is negative, we have complex solutions:

G dfrac{0 pm sqrt{-8}}{4}

G dfrac{pm 2isqrt{2}}{4}

G dfrac{1 pm isqrt{2}}{2}

4. Validate Solutions

Now, let's validate the solutions by substituting them back into the original equation. We have two potential solutions:

G dfrac{1 isqrt{2}}{2}

G dfrac{1 - isqrt{2}}{2}

4.1 First Solution

Substitute G dfrac{1 isqrt{2}}{2} into the original equation:

sqrt{-G1^2} Gsqrt{2}

sqrt{-(dfrac{1 isqrt{2}}{2})^2} dfrac{1 isqrt{2}}{2} sqrt{2}

sqrt{-(dfrac{1 2isqrt{2} - 2}{4})} dfrac{1 isqrt{2}}{2} sqrt{2}

sqrt{dfrac{-1 2isqrt{2}}{4}} dfrac{1 isqrt{2}}{2} sqrt{2}

dfrac{sqrt{-1 2isqrt{2}}}{2} dfrac{(1 isqrt{2})sqrt{2}}{2}

Note that the left-hand side and the right-hand side do not equate:

sqrt{-G1^2} ne Gsqrt{2}

4.2 Second Solution

Substitute G dfrac{1 - isqrt{2}}{2} into the original equation:

sqrt{-G1^2} Gsqrt{2}

sqrt{-(dfrac{1 - isqrt{2}}{2})^2} dfrac{1 - isqrt{2}}{2} sqrt{2}

sqrt{-(dfrac{1 - 2isqrt{2} - 2}{4})} dfrac{1 - isqrt{2}}{2} sqrt{2}

sqrt{dfrac{-1 - 2isqrt{2}}{4}} dfrac{(1 - isqrt{2})sqrt{2}}{2}

Note that the left-hand side and the right-hand side do not equate:

sqrt{-G1^2} ne Gsqrt{2}

5. Alternative Approach

We can also approach the equation using a different method. Consider the equation:

sqrt{-G1^2} Gsqrt{2}

sqrt{-G1^2} implies G1 can be positive or negative.

Let's consider iG1 Gsqrt{2}:

sqrt{-1} * G1 Gsqrt{2}

Implies i * G1 Gsqrt{2}

Implies G1 dfrac{Gsqrt{2}}{i}

Now rationalize the denominator:

G1 dfrac{Gsqrt{2}}{i} * dfrac{-i}{-i}

G1 dfrac{-iGsqrt{2}}{-1}

G1 iGsqrt{2}

Substitute G dfrac{1}{3} - isqrt{2}:

G1 i(dfrac{1}{3} - isqrt{2})sqrt{2}

G1 dfrac{i - 2isqrt{2}}{3}

Check:

sqrt{-G1^2} sqrt{-dfrac{i - 2isqrt{2}}{3}^2}

sqrt{dfrac{(i - 2isqrt{2})^2}{9}} sqrt{dfrac{-1 - 4sqrt{2}i - 2}{9}}

sqrt{dfrac{-3 - 4sqrt{2}i}{9}} dfrac{1 2isqrt{2}}{3}

After simplification:

G1 dfrac{1}{3} - isqrt{2}

The left-hand side and right-hand side are equal.

6. Conclusion

The solution to the equation sqrt{-G1^2} Gsqrt{2} is:

G dfrac{1}{3} - isqrt{2}

This solution satisfies both the original and the rationalized forms of the equation. Complex numbers can indeed provide solutions to seemingly intractable equations, showcasing the power of algebraic manipulation in advanced mathematics.