Implied Functions

 Original question

I'm trying to solve:

sqrt(a)+sqrt(b) = 0

Regardless of how I did manipulated this equation, I kept getting a=b.

sqrt(a) = -sqrt(b)

(sqrt(a))^2 = (-sqrt(b))^2

a=b

sqrt(a)+sqrt(b) = 0

If b = a = a*i^4, then:

sqrt(a)+sqrt(a*i^4) = 0

sqrt(a)+sqrt(i^4)*sqrt(a) = 0

sqrt(a)+i^2*sqrt(a) = 0

sqrt(a)-sqrt(a) = 0

Lo and behold... it worked. Not. So I decided to be a bit more creative and throw in some imaginary units.

sqrt(a) = -sqrt(b)

sqrt(a)/sqrt(b) = -1

sqrt(a/b) = -1

sqrt(a/b) = i^2

a/b = i^4

a = b*i^4

a - b*i^4 = 0

(sqrt(a)-sqrt(b)*i^2)*(sqrt(a)+sqrt(b)*i^2) = 0

(sqrt(a)+sqrt(b))*(sqrt(a)-sqrt(b)) = 0

In essence, the second part (bolded) was implied, if you can even call it that. Why do these implied functions pop out? Are they valid?

 Second question

sqrt(a) + sqrt(b) = c

sqrt(a) + sqrt(b*i^4) = c

sqrt(a) + sqrt(i^4)*sqrt(b) = c

sqrt(a) + i^2*sqrt(b) = c

sqrt(a) - sqrt(b) = c

Why does this process allow me to randomly change additions to subtractions? In fact, I could have ended with:

+/- sqrt(a) +/- sqrt(b) = c

Four different possibilities. Just by factoring 1 into i^4. Why? I don't understand. And I can reproduce this process with any function. Behold, the Pythagorean Theorem:

a^2 + b^2 = c^2

a^2 + sqrt(b^4) = c^2

a^2 + sqrt(b^4*i^4) = c^2

a^2 + sqrt(i^4)*sqrt(b^4) = c^2

a^2 + i^2*b^2 = c^2

a^2 - b^2 = c^2

I am mind blown.

Third question:

How do you solve:

sqrt(a) + sqrt(b) = -1

sqrt(a) + sqrt(b) + 1 = 0

(a^(1/4) + i*sqrt(sqrt(b)+1))*(a^(1/4) - i*sqrt(sqrt(b)+1)) = 0

a^(1/4) = +/- i*sqrt(sqrt(b)+1)

This is where it becomes very problematic. There's no way to get rid of that one fourth power without eliminating the +/-, which was exactly what I was trying to avoid to begin with.

In essence, my question is this: can the following ever exist, either as a real number or a complex number?

sqrt(a) = -1

What happens when something that doesn't exist arises? Imaginary numbers were made to handle sqrt(-1). Limits were made to handle indeterminate forms. So what about this?