# Solving complex number equations

Hello, Im trying to solve

(z^4-2+i)(z^2+1-i)=0

(z^2+1-i)=0

Does a=1, b=1 & c=-1?

IM meant to

(a) Give answers in polar form using the principal argument;
(b) Give answers in cartesian form

Cartesian is (x,y) is it not and polar is (r,theta) Im just not sure about the principal argument.

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LeonhardEuler
Gold Member
deryk said:
Does a=1, b=1 & c=-1?

No, a=1, b=0, and c=(1-i) because "a" is the coefficient of z^2, "b" is the coefficient of z and "c" is the constant term.

HallsofIvy
Homework Helper
deryk said:
Hello, Im trying to solve

(z^4-2+i)(z^2+1-i)=0

(z^2+1-i)=0

Does a=1, b=1 & c=-1?

IM meant to

(a) Give answers in polar form using the principal argument;
(b) Give answers in cartesian form

Cartesian is (x,y) is it not and polar is (r,theta) Im just not sure about the principal argument.
However, there is no reason to use the quadratic formula here- since
z2+ 1-i= 0, z2= -1+ i and you just take the square root (this is what the quadratic formula would give you anyway):
$$z= \sqrt{-1+ i}$$

You'll want to convert that into polar form to take the square root. The answer will then be in polar form and you will need to convert back to Cartesian form.
The polar form of a+ bi is re&theta;i where
$r= sqrt{a^2+ b^2}$ and $\theta= tan^{-1}(\frac{b}{a})$.

HallsofIvy said:
However, there is no reason to use the quadratic formula here- since
z2+ 1-i= 0, z2= -1+ i and you just take the square root (this is what the quadratic formula would give you anyway):
$$z= \sqrt{-1+ i}$$

You'll want to convert that into polar form to take the square root. The answer will then be in polar form and you will need to convert back to Cartesian form.
The polar form of a+ bi is reθi where
$r= sqrt{a^2+ b^2}$ and $\theta= tan^{-1}(\frac{b}{a})$.
How would you write z=+ or -(-1+i)^0.5 & z= + or - (2-i)^0.5 in polar and cartesian form? Thanks.

deryk said:
How would you write z=+ or -(-1+i)^0.5 & z= + or - (2-i)^0.5 in polar and cartesian form? Thanks.
Pretend the complex number under the radicand is a 2-dimensional position vector with components (-1,1) in the case of the first one. Get the argument (angle from the positive real axis) and modulus (magnitude of the vector) from that, then write it in polar form to get the roots.

HallsofIvy
Homework Helper
deryk said:
How would you write z=+ or -(-1+i)^0.5 & z= + or - (2-i)^0.5 in polar and cartesian form? Thanks.
Since you copied the formulas I gave for converting from cartesian to polar coordinates, I assume you are asking about the square root:

DeMoivre's formula: If a complex number has "polar coordinates" r, θ (in other words, the number can be written r(cos θ+ i sin θ) or re) then it's square roots are
$$r^{\frac{1}{2}}(cos\frac{\theta}{2}+ i sin\frac{\theta}{2})$$
and
$$r^{\rac{1}{2}}(cos(\frac{\theta}{2}+ \pi)+ i sin(\frac{\theta}{2}+\pi))$$
or
$$r^{\frac{1}{2}}e^{\frac{\theta}{2}}$$
and
$$r^{\frac{1}{2}}e^{(\frac{\theta}{2}+\pi)}$$