Mathematical simplicity

Mathematical simplicity.
Mathematics isn't always simple.
But sometimes, with a little thought, we can find very simple shortcuts, which are, however, surprisingly true.
We said that in quadratic functions f(x), for example, it was enough to change the sign of the monomial with an even exponent to obtain the point-symmetric function $ (for those who follow) called g(x).
This function g(x) has real roots if f(x) doesn't, having only two complex roots.
Now, what does [-b±sqrt(b²-4ac)]/2a become if we change the sign of a?
It becomes [-b±sqrt(b²+4ac)]/(-2a)
It's mathematical.
But that's not all, you'll remember, if you remember anything from what I said earlier: I said that the complex roots of a function are pure imaginaries, and that they are found by rotating f(x) 180° about the point $(0,y₀ ) to form g(x).
We then have here, directly, the complex roots of f(x), given in pure imaginaries, as should always be the case if we correctly understand what we are doing. Notations like x'=2+3i or x"=-3-i are a mathematical joke.
Real roots of quadratic functions: x= [-b±sqrt(b²-4ac)]/2a
Complex roots of quadratic functions:
x= {-[b±sqrt(b²+4ac)]/(2a)}.i
Beware of sign errors (the big trap of complex roots).
Example, let's set f(x)=x²+4x+5
x'=i
x"=-5i
If you replace x with i or -5i, you will get f(x)=0.
If you can't do this, it's because you haven't understood how imaginary numbers work, like 100% of the human beings on this earth.
Thank you for your attention.
R.H.