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Chapter 1: Problem 18

Find all complex numbers \(z\) such that $$ 4 z^{2}+8|z|^{2}=8 $$

Short Answer

Expert verified
Question: Find all the complex numbers z that satisfy the equation \(4z^2 + 8|z|^2 = 8\). Answer: The complex numbers z that satisfy the equation are: \(\{ \sqrt{\frac{2}{3}}, -\sqrt{\frac{2}{3}}, \sqrt{2}i, -\sqrt{2}i\}\).

Step by step solution

01

Represent z in terms of its real and imaginary parts

Let z = a + bi, where a and b are real numbers and i is the imaginary unit. Then, the equation we are given can be written as: $$ 4(a+bi)^2 + 8(a^2 +b^2) = 8 $$
02

Separate the real and imaginary parts

Now we will expand and separate the real and imaginary parts of the equation: $$ 4(a^2 + 2abi - b^2) + 8(a^2 +b^2) = 8 $$ Separating the real and imaginary parts, we have: $$ 4(a^2-b^2)+8(a^2+b^2)=8 \\ 8ab = 0 $$
03

Solve the real equations

We have two equations to solve for a and b: $$ 4(a^2-b^2)+8(a^2+b^2)=8 \\ 8ab = 0 $$ From the second equation, we find that either a = 0 or b = 0. Case 1: a = 0 $$ 4(-b^2)+8b^2 = 8 \\ 4b^2 = 8 \\ b^2 = 2 \\ b = \pm\sqrt{2} $$ So, when a = 0, we have z = 0 + \sqrt{2}i and z = 0 - \sqrt{2}i. Case 2: b = 0 $$ 4(a^2)+8a^2 = 8 \\ 12a^2 = 8 \\ a^2 = \frac{2}{3} \\ a = \pm\sqrt{\frac{2}{3}} $$ So, when b = 0, we have z = \sqrt{\frac{2}{3}} + 0i and z = -\sqrt{\frac{2}{3}} + 0i.
04

Combine the solutions

Combining the solutions from both cases, we find that the complex numbers z that satisfy the given equation are: \(\{ \sqrt{\frac{2}{3}}, -\sqrt{\frac{2}{3}}, \sqrt{2}i, -\sqrt{2}i\}\).

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Most popular questions from this chapter

Find all complex numbers \(z \neq 0\) such that \(z+\frac{1}{z} \in \mathbb{R}\).

Find all positive integers \(n\) such that $$ \left(\frac{-1+i \sqrt{3}}{2}\right)^{n}+\left(\frac{-1-i \sqrt{3}}{2}\right)^{n}=2 $$

Solve the following equations: (a) \(z \cdot(1,2)=(-1,3) ;\) (b) \((1,1) \cdot z^{2}=(-1,7)\).

Consider the complex numbers \(z_{1}=(1,2), z_{2}=(-2,3)\), and \(z_{3}=\) \((1,-1)\). Compute the following: (a) \(z_{1}+z_{2}+z_{3}\) (b) \(z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\); (c) \(z_{1} z_{2} z_{3} ;\) (d) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}\) (e) \(\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}\) (f) \(\frac{z_{1}^{2}+z_{2}^{2}}{z_{2}^{2}+z_{3}^{2}}\)

Find the set of points \(P(x, y)\) in the complex plane such that $$ \left|\sqrt{x^{2}+4}+i \sqrt{y-4}\right|=\sqrt{10} $$
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