Finally, but it took much more than half an hour LOL. That's because we're not in the same room talking.
I looked at what you wrote and I don't see a problem with it initially, except you put a b^2 where it didn't belong, but that doesn't matter to me because it did not affect your final result.
x = [b' +- sqrt(b'^2 + 4ac')]/[2a] = [(-b)^2 +- sqrt((-b)^2 + 4a(-c))]/[2a]
The rest of it seems fine to me but I must caution you, I am not as rigorous as others. To me, it seems a tiny bit like circular reasoning but I am no expert in logic. Let others more qualified make their assessment of your proof.
I wanted to avoid any connection to the textbook version, that's why i considered the 4 forms, made my arguments, definitions, and derivations.
If i had an equation to solve in the form ax^2 + bx + c = 0 Then I would use the textbook version.
If you gave me 10 equations to solve in the form ax^2 = bx + c then I would feel silly using the textbook version because i know a better version for this form. My version. Imagine... sitting there with pencil and paper, collecting terms on one side, carrying around un-necessary minus signs, transforming subtractions to additions, squaring negatives, on and on, for 10 equations. Or better yet, ask a professional mathematician, who doesn't know, or won't accept my version because it's 'trivial', to do 10 equations by hand, would you chuckle as you watch him struggle?
If you were throwing random forms at me, I wouldn't know what i was going to get next out of possible
form1 ax^2 + bx + c = 0
form2 ax^2 + bx = c
form3 ax^2 + c = bx
form4 ax^2 = bx + c
And you asked me to solve 100 random forms or 1000 or 10^6 random 2nd degree equations, Then I would use my version because I would expect 75% of the forms thrown at me would not be form1
Don't forget that the textbook version has an extra minus sign and subtraction instead of addition. Those 2 differences are going to cause more problems. Does that make sense?
Thank you for your reply, thank you for your comments.