# How do you find all the complex roots of 5x^6 + x^4 -3 = 0?

Feb 12, 2016

Solve as a cubic equation in $\frac{1}{x} ^ 2$ using Cardano's method.

#### Explanation:

Let $t = \frac{1}{x} ^ 2$

Then our polynomial equation becomes: $\frac{5}{t} ^ 3 + \frac{1}{t} ^ 2 - 3 = 0$

Multiply through by ${t}^{3}$ and rearrange to get:

$3 {t}^{3} - t - 5 = 0$

Use Cardano's method:

Let $t = u + v$

Then:

$0 = 3 {\left(u + v\right)}^{3} - \left(u + v\right) - 5$

$= 3 {u}^{3} + 3 {v}^{3} + \left(9 u v - 1\right) \left(u + v\right) - 5$

Add the constraint $v = \frac{1}{9 u}$ (causing $\left(9 u v - 1\right) = 0$) to get:

$= 3 {u}^{3} + \frac{1}{243 {u}^{3}} - 5$

Multiply through by $243 {u}^{3}$ to get a quadratic equation in ${u}^{3}$:

$729 {\left({u}^{3}\right)}^{2} - 1215 \left({u}^{3}\right) + 1 = 0$

This has roots given by the quadratic formula:

${u}^{3} = \frac{1215 \pm \sqrt{{1215}^{2} - \left(4 \cdot 729\right)}}{1458}$

$= \frac{1215 \pm \sqrt{1476225 - 2916}}{1458}$

$= \frac{1215 \pm \sqrt{1473309}}{1458}$

$= \frac{1215 \pm 27 \sqrt{2021}}{1458}$

$= \frac{45 \pm \sqrt{2021}}{54}$

The derivation was symmetric in $u$ and $v$, so one of these roots is ${u}^{3}$ and the other ${v}^{3}$. Hence the Real root of our cubic in $t$ is:

${t}_{1} = \sqrt[3]{\frac{45 + \sqrt{2021}}{54}} + \sqrt[3]{\frac{45 - \sqrt{2021}}{54}}$

$= \frac{1}{3} \sqrt[3]{\frac{45 + \sqrt{2021}}{2}} + \frac{1}{3} \sqrt[3]{\frac{45 - \sqrt{2021}}{2}}$

and the non-Real Complex roots are given by:

${t}_{2} = \frac{1}{3} \omega \sqrt[3]{\frac{45 + \sqrt{2021}}{2}} + \frac{1}{3} {\omega}^{2} \sqrt[3]{\frac{45 - \sqrt{2021}}{2}}$

${t}_{3} = \frac{1}{3} {\omega}^{2} \sqrt[3]{\frac{45 + \sqrt{2021}}{2}} + \frac{1}{3} \omega \sqrt[3]{\frac{45 - \sqrt{2021}}{2}}$

where $\omega = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ is the primitive Complex cube root of $1$

Hence the roots of our original equation are:

${x}_{1 , 2} = \pm \frac{1}{\sqrt{{t}_{1}}}$

${x}_{3 , 4} = \pm \frac{1}{\sqrt{{t}_{2}}}$

${x}_{5 , 6} = \pm \frac{1}{\sqrt{{t}_{3}}}$

For example,

x_3 = 1/sqrt(1/3 omega root(3)((45+sqrt(2021))/2) + 1/3 omega^2 root(3)((45-sqrt(2021))/2)

I have previously explained how to get the square root of a Complex number in $a + b i$ form, which you can use to help get these roots into $a + b i$ form.