# Tagged Questions

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### When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn. For any four complex numbers $a$, $b$, $c$, $d$, the following identity is easy ...
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### Function with exponent imaginary power

If we have $u=\frac{4c(e^{-is}-e^{is})}{(e^{-is}+e^{is})^2} \tag 1$ where c is a constant and s is a variable. Can we write $e^{is}$ in terms of u ? Means Can we write $e^{is}$ as $\psi(u)$ , a ...
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### Find all complex and real roots of higher degree polynomials, given one root

$2+3i$ is a zero of $f(x)=x^4-4x^3+17x^2-16x+52$, find all of the zeros of $f(x)$ thanks!
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### Roots of Unity - $x^3 = -i$

I need to find the roots of unity for the following equation: $$x^3 + i = 0$$ Thus, $x^3 = -i$. I know that $-i = \exp[i(\frac{3\pi}{2} + 2n \pi)]$ however I do not know how to get all roots. ...
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### Find all values for cos(i)

In my Differential Equations class recently we have learned about Euler's Formula and Fourier Series. I am given the problem ...
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### Third point of a triangle in the complex plane

I have an equilateral triangle with two points equal to $(2+2i)$ and $(5+i)$. I want to find the third point(s) (there are $2$ of these). I have that the side length of the triangle is $\sqrt{10}$.
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### Complex numbers and geometry

There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.
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### Complex solutions to $x^3 + 512 = 0$

An algebra book has the exercise $$x^3 + 512 = 0$$ I can find the real solution easily enough with $$x^3 = -512$$ $$\sqrt[3]{x^3} = \sqrt[3]{-512}$$ $$x = -8$$ The book also gives the ...
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### Pre-calc complex number geometry

The equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38$ for some complex numbers a and b. Find the product $ab$. Maybe ...
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### Find all complex solutions to the equation

i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0 I basically have no clue, any tips/advice/solutions would be great. I could also need some help with another question, this one ...
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### quotient of complex numbers?

so I was wondering if you have two different equations having denominators $2+i$ and $2-i$ respectively how came the denominator of the quotient in standard form is $5$ for both equations? I tought ...
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### How to solve this equation? $8z^3+1=0.$ [closed]

I would like to know how to do this question. (a) Determine all of the solutions of the equation $$8z^3+1=0.$$ Express your solutions in the form $z = re^{i\theta}$ where $-\pi<\theta \leq \pi$. ...
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### How to solve the complex equation? $(x+2yi)^2 = xi.$

How to solve the following complex equation with in less than 60 seconds? $$(x+2yi)^2 = xi.$$ I know how to solve, we have to solve power first then real part equal to real part and imaginary to ...
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### Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
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So, I am computing something seemingly simple involving complex gaussians and constants, but I am getting a big contradiction in my calculations. The setup: Let $C$ be a complex constant, that is, ...
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### Solving two varibles system equation above $\mathbb{C}$

A bit emmbarrassed to ask this newbie question: Let: $$(1+i)x + y = 2$$ $$(1-i)x + iy = 0$$ Multiplying the first equation by $(-i)$ and summing the two equations, we have: $$(2-2i)x + 2i = 0$$ ...
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### Prove that function transforms $S$ into $S$

I have to prove, that function $\Phi(z):=\frac{1-\overline a z}{z-a}$ transforms $S$ into $S$ where $S:=\{|z|=1\}$ and $|a|>1$ I don't know where can I start. Any hint?
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### What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
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### Calculating real and imaginary part of a complex number

Consider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$. How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these ...
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### How to teach newbie multiply of complex number

I want to teach a newbie the arithmetic law of complex numbers. the law of add is acceptable psychological. but multiply is not. for example, assume $$z = a+bi, w = c+di$$ He (She) may ask me: why ...
### What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?
$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$