Problem 43 Solve for all possible values of... [FREE SOLUTION]

Chapter 2: Problem 43

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

Short Answer

Expert verified

The solutions are $ x = 0, y = 0 $, $ x = 1, y = 1 $, $ x = -1, y = 1 $.

Step by step solution

Expand the left side

First, expand the left side of the equation \ \ (x + i y)^{2}. \ Use the distributive property to get \ \[ (x + i y)^{2} = x^{2} + 2 i x y - y^{2} \] \

Compare real and imaginary parts

Write the expanded form next to the right-hand side \ \[ x^{2} + 2ixy - y^{2} = 2ix \] Equate the real parts and the imaginary parts separately. \ Real part: \ $ x^2 - y^2 = 0 $ \ Imaginary part: \ $ 2xy = 2x $.

Solve the real part

Solve the equation for the real parts \ $ x^2 - y^2 = 0 $. \ This can be rewritten as $ x^2 = y^2 $. \ Taking the square root of both sides, we get \ $ x = y $ or $ x = -y $.

Solve the imaginary part

Now solve the equation for the imaginary parts \ \ $ 2xy = 2x $. \ Divide both sides by 2: \ \ $ xy = x $. \ This can be written as \ \ $ x(y - 1) = 0 $. \ Thus, $ x = 0 $ or $ y = 1 $.

Combine solutions

Combine the solutions from Steps 3 and 4. If $ x = 0 $, we substitute into $x = y$ or $x = -y$: \ \ $ x = 0, y = 0 $ \ If $y = 1$, substitute into $ x = y $ or $ x = -y $: \ $ x = 1, y = 1 $ or $ x = -1, y = 1 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers

Complex numbers extend the concept of one-dimensional real numbers to two-dimensional numbers. A complex number is written in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part multiplied by $i$ (the imaginary unit, which is defined as $i^2 = -1$). Complex numbers allow us to solve equations that have no real solutions and are crucial in fields such as engineering and physics.
In our exercise, $x + i y$ is a complex number where \[x\] is the real part and \[y\] is the imaginary part.

Real and Imaginary Parts

In a complex number $a + bi$, the term $a$ is the real part and $bi$ is the imaginary part. To solve complex number equations, we often need to separate these parts.
In our exercise, after expanding $ (x + i y)^{2} $, we compare the real and imaginary parts separately:
$x^2 - y^2 = 0$ is the equation for the real part and $2ixy = 2ix$ is for the imaginary part. By treating these separately, we can solve for the individual variables $x$ and $y$.

Algebraic Solutions

Algebraic solutions involve manipulating algebraic expressions to solve for unknown variables. For the real part, $x^2 - y^2 = 0$ signifies that either $x = y$ or $x = -y$. For the imaginary part, we have $2xy = 2x$. By dividing both sides by 2, we arrive at $xy = x$, which simplifies to \[x(y-1) = 0\]. This implies either $x=0$ or $y=1$. Combining these provides the complete set of solutions.

Quadratic Equations

Quadratic equations take the form \[ax^2 + bx + c = 0\]. The quadratic equation in our exercise is hidden in the form $(x + i y)^2 = 2ix$. Upon expanding and isolating real and imaginary parts, we solve the resulting simpler quadratic equations.
In particular, solving $x^2 - y^2 = 0$ gives $x = y$ or $x = -y$ as it forces us to find specific values of $x$ and $y$ that satisfy the original complex equation.

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