Roots of a second degree equation.

[Richard Hachel]

Roots of a quadratic equation.

y=ax²+bx+c

If b² >4ac then there are two roots.

If b=4ac then there is a single root.

If b²<4ac there are no roots.

We can then draw as many curves as we want, as long as we want, nothing
will change, there are no roots.

At least in the real world.

Let us set y=x²+1 or y=x²+4x+5; there are no roots.

This does not exist, looking for roots in nothingness, or rabbit horns
will not change anything. We will not find any.

Some mathematicians will then try to find some anyway, but beyond reality,
where Doctor Hachel takes possession of your computer screen and will give
it back to you only if he wants to (I put a virus in the equation
mentioned above).

They call these imaginary roots, because, since they do not exist, we must
imagine them. But what do they correspond to?

It would seem, in fact, that they are not the roots of the equation, that
is to say the place in x where the curves cross y=0 since it is impossible
for all x, but the horizontal mirror projection passing through the vertex
of the curve.

For y=x²+1 then [-b(+-)sqrt(b²-4ac)]/2a --->
(+-)sqrt(-4ac)/2
(+-)sqrt(4i²)/2=(+-)i

The two roots are x'=-i and x'=i (i.e. x=-1 and x=1 of the imaginary
inverted curve).

Let us set =x²+4x+5 which has no root, and project this curve in mirror;
two roots appear for the mirror curve.
[-b(+-)sqrt(b²-4ac)]/2a --->[-4(+-)sqrt(16-4*5)]/2 --->
[-4(+-)sqrt(4i²)]/2a--->(-4(+-)2i)/2
x=-2-i
x'=-3
x"=-1

But this is very interesting, but wouldn't it be worth giving right away
the equation of the mirror curve whose vertex touches the vertex of the
real curve by specifying that it is the imaginary horizontal mirror?

This means that a second degree curve has two roots (or a single one) and
that when it doesn't have any, what crosses y=0 is its imaginary mirror
curve which will give two imaginary roots.

R.H.

[Moebius]

What exactly do you mean by saying that the imaginary roots (usually
called complex roots by mathematicians) do not exist? What attribute
does -2 + i possess, or lack, that entitles you to attribute to it the
property of non-existence? How does -2 + i differ in that respect from
other numbers such as -1 or 42?
The fact is, there is a vast theory of complex analysis which is
coherent and fascinating. It is also useful in science and engineering.

Even "worse":

See: https://www.nature.com/articles/s41586-021-04160-4

[Alan Mackenzie]

Post by Richard Hachel
Roots of a quadratic equation.
y=ax²+bx+c
If b² >4ac then there are two roots.
If b=4ac then there is a single root.
If b²<4ac there are no roots.
We can then draw as many curves as we want, as long as we want, nothing
will change, there are no roots.
At least in the real world.
Let us set y=x²+1 or y=x²+4x+5; there are no roots.
This does not exist, looking for roots in nothingness, or rabbit horns
will not change anything. We will not find any.
Some mathematicians will then try to find some anyway, but beyond reality,
where Doctor Hachel takes possession of your computer screen and will give
it back to you only if he wants to (I put a virus in the equation
mentioned above).
They call these imaginary roots, because, since they do not exist, we must
imagine them. But what do they correspond to?

What exactly do you mean by saying that the imaginary roots (usually
called complex roots by mathematicians) do not exist? What attribute
does -2 + i possess, or lack, that entitles you to attribute to it the
property of non-existence? How does -2 + i differ in that respect from
other numbers such as -1 or 42?

The fact is, there is a vast theory of complex analysis which is
coherent and fascinating. It is also useful in science and engineering.

[ .... ]

Post by Richard Hachel
R.H.