Complex numbers - homework question, please help ASAP

[astonky]

Hi,

I have two homework questions on complex numbers that I don't understand how to do. Could you please show me the exact steps (I have more similar questions, so I can practise regardlessly) and why you do each step. Thank you very much.

 

Here are the questions:

Find the complex number z that satisfies these equations:

(z+1)i = (z+2i)(3+2i)

Solve for z is a Caresian number:

(modulus of z) -z = 4+3i

[kw0573]

Usually when we solve, we assume z = a + bi, where a and b are real numbers.

Then usually, expand the brackets, move constants to one side of equation and variables to other, and solve for real and imaginary parts separately. 

(a + bi + 1)i = (a + bi + 2i)(3 + 2i)

ai - b + i = 3a + 2ai + 3bi - 2b + 6i - 4 (multiply out the brackets)

Now get all the a's and b's to the right, other terms to the left. Separate real and imaginary terms. For example since a and b are real numbers, only terms with "i" are imaginary.

4 - 6i + i = 3a - 2b + b + 2ai - ai + 3bi

Now the step that justifies why we bothered to create new variables a and b is here, where we match real and imaginary parts from both sides
Equation 1:...... 4 = 3a - 2b + b = 3a - b

Equation 2:...... (-5)i = (2a - a + 3b)i ==> -5 = a + 3b

Solve this system of equations to get a = 7/10, b = -19/10, z = 7/10 - 19i/10

[kw0573]

For second question, also assume z = a + bi
We have |z| - z = 4 + 3i, note that |z| is a real number, after rearranging we get
|z| - 4 - 3i = z = a + bi
By matching we see
|z| - 4 = a, and -3 = b
expand |z| to be sqrt(a^2 + b^2), we know b = -3, so |z| = sqrt(a^2 + 9)

sqrt(a^2 + 9) - 4 = a --> sqrt(a^2 + 9) = a + 4, square both sides
a^2 + 9 = a^2 + 8a + 16

9 = 8a + 16

-7 = 8a ---> a = -7/8

z = -7/8 - 3i

[astonky]

Wow, thanks thats great you helpex me so much!

[haese225]

Q: Find the complex number z

(z + 1)i = (z + 2i)*(3 + 2i)

Since its linear here's another way to solve for z:

Multiplying in i to both sides,

iz + i = z(3 + 2i) +6i - 4

Rearrange for z

4 - 5i = z(3 + i)

z = (4 - 5i)/(3 + i)

Multiply 3 - i (conjugate roots technique)

z = (12 - 4i - 15i - 5)/10

z = (7 - 19i)/10

A bit late to reply but at least its a quick way?

Edit: Missing (-) added...

Edited September 25, 2016 by 4lan