# Need help solving an quartic equation.

#### Student4

Hi.

1. Homework Statement

Find the four roots of:
https://www.physicsforums.com/attachment.php?attachmentid=39371&stc=1&d=1317233718

2. Homework Equations
Calculus: p.6 Polynomials and Rational functions

3. The Attempt at a Solution
Pretty much stuck on starting it. I can't really find a way to start factoring it.

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#### spamiam

Try using a substitution to turn it into a quadratic.

#### Student4

hi spamiam, thanks for fast reply. Could you just show me fast how to get this one down to quadratic, only if you have time ofc. Have done similar equations with substitution, but this spesific one i can't get my head around.

#### dynamicsolo

Homework Helper
Concerning spamiam's suggestion, keep in mind that z4 can be written as ( z2 )2 ...

#### SammyS

Staff Emeritus
Homework Helper
Gold Member
Let u = z2 or u = z2 - 1

#### Student4

so;

(z²)²-2z²+4=0 and let u=z²

u²-2u+4=0

u= 1+i√3 and u= 1-i√3

And i find z

z²= 1+i√3 and z²= 1-i√3

z= √(1+i√3), z= -√(1+i√3), z= √(1-i√3), z=√(1-i√3)

is this correct? i think the answer shud be more simplified

#### dynamicsolo

Homework Helper
If you write the result in Cartesian form (a + ib), you won't be able to "simplify" this much. Do you know how to write complex numbers in polar form? You could then get "simpler" expressions using DeMoivre's Theorem.

#### spamiam

dynamicsolo is right, polar form is much cleaner, especially if you note that $\frac{1\pm i\sqrt{3}}{2}$ are primitive sixth roots of unity and can be written $e^{\frac{2 \pi i}{6}}$ and $e^{2 \pi i\frac{5}{6}}$.

#### Student4

Hi, dynamicsolo, yes shud know it, but just started my course so im abit new to polar form. I need to find modulus and argument of the complex number first right?

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#### spamiam

Complex numbers can be depicted as points in the plane. Just draw yourself a triangle: the real axis is horizontal, and the imaginary is vertical. This gives you a triangle with what side lengths? Then the modulus is just the length of the hypotenuse, and the argument is the angle between the hypotenuse and the horizontal axis.

#### epenguin

Homework Helper
Gold Member
But for this problem #2 and #4 have as near told you what to do as anyone could without writing out your work for you.

#### Student4

Still abit confused

First root z= √(1+i√3)
simplifed with polarform and DeMoivre's Theorem:

modulus: √(1^2 + √3^2 ) = 2
argument: tan = √3/1 = pi/3

and polarform w=( 2*(cos(pi/3) + i sin(pi/3) )^1/2 ,and then with the theorem i can write: √2 (cos(1/2*pi/3) + i sin(1/2*pi/3))

And then i get: √6/2 + i*√2/2 as my first root.

but for next root z= -√(1+i√3) i have a problem. What do i do when i have -√?

either i plug that in at the end and get same as first root just -(root 1)

or i switch operator z= -√(1+i√3) -> z= √(1-i√3), but then i get same root as my 3th root z= √(1-i√3)

Any tip what to do

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