Question

Solve for all possible values of the real numbersand in the following equations ${\left(x-iy\right)}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathit{i}\mathit{x}$.

Short answer

The answer is $\left(x,y\right)=\left(0,0\right)or\left(1,1\right)or\left(-1,1\right).$

Step-by-step solution

Given information

It is given that${\left(x-iy\right)}^2=2ix.$ .

Definition of a complex number

Every complex number can be described as:

z=a+bi

Where aand bare both real numbers, z is the complex number, and i is known as iota, which makes z a complex number.

Begin by stating the given information

State the given information as:

$$\begin{gathered} {\left(x-iy\right)}^2=2ix \\ {x}^2+2xyi-y^2=2xi \end{gathered}$$

Equate like terms

Equate like terms from both sides:

$$\begin{gathered} x^2-y^2=0 \\ {x}^2=y^2 \\ and \\ x\left(y-1\right)=0 \end{gathered}$$

Evaluate possible solutions

The above equation implies:

$$\begin{gathered} Ifx=0theny=0 \\ Ify=1thenx=\pm 1 \end{gathered}$$

Thus, the final answer is $\left(x,y\right)=\left(0,0\right)or\left(1,1\right)or\left(-1,1\right)$.