### NORMALIZED CUBIC EQUATION ROOTS

NORMALIZED CUBIC EQUATION ROOTS  .

.

```
navy    Normalized Cubic Equation

The plot shows the roots of the cubic equation in one diagram.
It serves as the general case,

${z}_{0}$,

by using a linear transformation
for both the independent and dependent variable.  It should be
useful for rough calculations or starting points for Newton's
method.  Explicit formulas for the real and imaginary component
of each root are given so the ambiguity of root choice is not left

Normalized Cubic Equation   The single real root is blue
Analytical expressions for the curves are developed at
cubic_exp.xml  from
the expressions reported at
Weisstein
wikipedia  and by
Knaust
The real and imaginary components of the derived expressions were found
to be exact fits.

Exact agreement for the first real root when all roots are real.

$z=\sqrt{\frac{n}{2}}\bullet \sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}}\bullet 2\bullet \mathrm{cos}\left(\frac{0\bullet \pi }{6}+\frac{\mathrm{arctan}\left(\sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}-1}\right)}{3}\right)-\frac{1}{3}$

The imaginary components of the complex roots are red color coded curves.

The real component of the complex roots is color coded orange.

The real component of the complex roots is color coded orange.

fuchsia

magenta

pink

green

$z=\sqrt{\frac{n}{2}}\bullet \sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}}\bullet \mathrm{cos}\left(\frac{0\bullet \pi }{6}+\frac{\mathrm{arctan}\left(\sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}-1}\right)}{3}\right)-\frac{1}{3}±\sqrt{\frac{n}{2}}\bullet \sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}}\bullet \sqrt{3}\bullet \mathrm{sin}\left(\frac{0\bullet \pi }{6}+\frac{\mathrm{arctan}\left(\sqrt{\frac{4\bullet {m}^{3}}{{n}^{2}}-1}\right)}{3}\right)$

maroon

For an angle of
pi/7
the
solution
is transcendental
but for pi/60 the solution is only irrational

$2\bullet \mathrm{cos}\left(2\bullet \pi /7\right)={\left(\frac{7}{\mathrm{27}}\bullet \frac{\sqrt{1+{3}^{3}}}{2}\right)}^{\left(1/3\right)}2\bullet \mathrm{sin}\left(\frac{\pi -\frac{2}{3}\mathrm{atan}\left(\sqrt{\mathrm{27}}\right)}{2}\right)-\frac{1}{3}$

cyan
Khayyam, Omar, 1048-1131
studied the intersection of conic sections and found 14 forms of the cubic
equation and all their branches.  He found that the right triangle problem
with the hypotenuse equal to a leg plus the altitude resulted in a cubic
equation.

darkorange

If you do not have the .svg and MathML plugins this page will render

plugin detection page  to find out what needs to be done.
Another

involved graph
illustrates voltage, current, power and
impedance relationships of regulations.
There are graphical computational aids and calculators at

www.ConcurrentInverse.com .  There are

caveats to our unbrideled technology  and

overzealous autocrats.

olive

Wm. C.

Corwin

```