Question

Solve for all possible values of the real numbers \(x\) and \(y\) in the following equations. $$(x+i y)^{2}=(x-i y)^{2}$$

Short answer

The solutions are \( (x=0, y \text{ any real}) \ and \ (y=0, x \text{ any real}) \.

Step-by-step solution

Expand both sides using the binomial formula

First, expand \((x+iy)^{2} \) and \((x-iy)^{2}\) using the binomial formula \((a+b)^{2}=a^{2}+2ab+b^{2}\).

Simplify the expanded forms

For \((x+i y)^{2}\), the expanded form is \ x^{2} + 2i x y - y^{2} \. For \ (x-i y)^{2} \, the expanded form is \x^{2} - 2i x y - y^{2} \.

Equate the real and imaginary parts

Set the real part of the left side equal to the real part of the right side, and do the same for the imaginary parts: \( x^{2} - y^{2} = x^{2} - y^{2} \) (this is always true) and \( 2ixy = -2ixy \).

Solve for the imaginary part equation

From \( 2ixy = -2ixy \), we obtain \( 4ixy = 0 \). This leads to \(xy = 0 \). Either \x = 0 \ or \ y = 0 \.

Find all possible solutions

Combine the results from the previous step to list all possible pairs \( x, y \). These are \ (x=0, y \text{ any real number}) \ and \ (y=0, x \text{ any real number}) \.

Key concepts

Binomial Expansion

The binomial expansion is a crucial method in solving algebraic equations, including those involving complex numbers. It involves expanding expressions raised to a power, usually in the form \((a + b)^n\). For instance, for \((x + iy)^2\), applying the binomial formula \((a + b)^2 = a^2 + 2ab + b^2\), we get:
- First term: \((x)^2 = x^2\)
- Second term: \((2 * x * iy) = 2ixy\)
- Third term: \((iy)^2 = i^2 y^2 = -y^2\) (since \(i^2 = -1\))
Therefore, the expansion results in \((x^2 + 2ixy - y^2)\). Similarly, \((x - iy)^2\) can be expanded to \((x^2 - 2ixy - y^2)\). This expansion is the first step in solving complex number equations as it transitions the problem into more manageable real and imaginary components.

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is key when dealing with complex equations. A complex number is generally expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. In the complex number equation \((x + iy)^2 = (x - iy)^2\), once expanded, each side of the equation can be separated into its real and imaginary components:
- Real part of \((x + iy)^2\): \(x^2 - y^2\)
- Imaginary part of \((x + iy)^2\): \(+2ixy\)
- Real part of \((x - iy)^2\): \(x^2 - y^2\)
- Imaginary part of \((x - iy)^2\): \(-2ixy\)
By equating the real parts together and the imaginary parts together, we can simplify and solve the complex equation step-by-step.

Solving Complex Equations

Solving complex number equations often involves transforming the complex numbers into their real and imaginary parts, then setting equational components equal to each other. In the given problem, after expanding both sides, we equate:
1. The real parts: \(x^2 - y^2 = x^2 - y^2\). Since the equation is always true, it gives us no new information.
2. The imaginary parts: \(2ixy = -2ixy\). Simplifying leads to \4ixy = 0\. This results in \xy = 0\.
From here, we solve for \x\ and \y\ by exploring the values that satisfy \xy = 0\; namely, either \x = 0\ or \y = 0\. Both scenarios give us possible solutions.

Real Numbers

To finalize the complex equation solutions, we need to understand real numbers \(x\) and \(y\):
- If \(x = 0\), then \(y\) can be any real number, leading to the solution set: \((x = 0, y \text{ any real number})\).
- If \(y = 0\), then \(x\) can be any real number, leading to the solution set: \((y = 0, x \text{ any real number})\).
Thus, the complete solutions for the equation \((x + iy)^2 = (x - iy)^2\) are derived by combining these results: \( (0, y) \text{ for any real } y \text{ and } (x, 0) \text{ for any real } x\).
Remember, separating variables into their real and imaginary parts makes solving complex equations easier and provides clear insights into the structure of the problem.