Simplify Complex Expression: (x+2i)(3-2xi)+2x^2i

When dealing with complex numbers, you often encounter expressions that need to be simplified into a standard form, typically $a+bi$, where $a$ is the real part and $b$ is the imaginary part. This article will guide you through simplifying the expression $(x+2 i)(3-2 x i)+2 x^2 i$, where $x$ is a real number. We'll break down each step, ensuring you understand how to combine real and imaginary components to arrive at the simplest form. This process involves the distributive property (often referred to as FOIL for binomials), careful handling of the imaginary unit $i$ (remembering that $i^2 = -1$), and combining like terms. By the end, you'll be able to confidently tackle similar complex number simplification problems. Let's dive into the specifics of simplifying this particular expression, ensuring that every step is clear and easy to follow.

Understanding the Basics of Complex Numbers

Before we get into the nitty-gritty of the simplification, let's quickly recap what complex numbers are and the rules we'll be using. A complex number is generally expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. The imaginary unit $i$ is defined as the square root of -1, so $i = \sqrt{-1}$, which means $i^2 = -1$. This property is crucial for simplifying expressions involving $i^2$. When we multiply complex numbers, we use the distributive property, similar to multiplying binomials. For example, $(a+bi)(c+di) = ac + adi + bci + bdi^2$. Since $i^2 = -1$, this becomes $ac + adi + bci - bd$, which can then be rearranged into the standard $a+bi$ form by grouping the real terms ($ac-bd$) and the imaginary terms ($ad+bc$) together: $(ac-bd) + (ad+bc)i$. We'll be applying these fundamental principles to our given expression $(x+2 i)(3-2 x i)+2 x^2 i$. Understanding these foundational concepts is key to successfully navigating the simplification process and ensuring accuracy in your final answer. It's always a good idea to double-check your understanding of these rules before proceeding with more complex calculations, as a solid grasp of the basics will prevent errors down the line.

Step-by-Step Simplification of the Expression

Let's begin by tackling the first part of the expression: $(x+2 i)(3-2 x i)$. We will use the distributive property (FOIL) to expand this product:

• First: $x \times 3 = 3x$
• Outer: $x \times (-2xi) = -2x^2i$
• Inner: $2i \times 3 = 6i$
• Last: $2i \times (-2xi) = -4x i^2$

So, $(x+2 i)(3-2 x i) = 3x - 2x^2i + 6i - 4xi^2$.

Now, we need to address the $i^2$ term. Remember that $i^2 = -1$. Substituting this into our expanded expression:

$3x - 2x^2i + 6i - 4x(-1)$

$3x - 2x^2i + 6i + 4x$

Next, we group the real terms and the imaginary terms. The real terms are $3x$ and $4x$. The imaginary terms are $-2x^2i$ and $6i$.

Combining the real terms: $3x + 4x = 7x$

Combining the imaginary terms: $-2x^2i + 6i = (-2x^2 + 6)i$

So, the simplified form of $(x+2 i)(3-2 x i)$ is $7x + (-2x^2 + 6)i$.

Now, let's incorporate the second part of the original expression, which is $+2x^2i$. We add this to our simplified result:

$(7x + (-2x^2 + 6)i) + 2x^2i$

To get the final $a+bi$ form, we combine the real parts and the imaginary parts.

• Real Part: The only real term is $7x$. So, $a = 7x$.
• Imaginary Part: The imaginary terms are $(-2x^2 + 6)i$ and $2x^2i$. Combining these gives us: $(-2x^2 + 6)i + 2x^2i = (-2x^2 + 6 + 2x^2)i$

Let's simplify the coefficient of $i$: $-2x^2 + 6 + 2x^2$. The $-2x^2$ and $+2x^2$ terms cancel each other out, leaving just $6$. So, the imaginary part is $6i$. This means $b=6$.

Therefore, the expression $(x+2 i)(3-2 x i)+2 x^2 i$ in its simplest $a+bi$ form is $7x + 6i$. This step-by-step approach ensures that no part of the original expression is overlooked and that all terms are correctly combined according to the rules of complex arithmetic. The careful separation of real and imaginary components throughout the process is what allows us to arrive at the neat final answer.

Conclusion and Further Exploration

In summary, we have successfully simplified the complex expression $(x+2 i)(3-2 x i)+2 x^2 i$ to its simplest $a+bi$ form. By applying the distributive property, remembering that $i^2 = -1$, and carefully grouping like terms, we arrived at the result $7x + 6i$. This expression is now in the standard form where $a=7x$ and $b=6$. This exercise highlights the importance of meticulous algebraic manipulation when working with complex numbers. The cancellation of the $x^2$ terms in the imaginary part was a key simplification that made the final answer particularly elegant.

Complex numbers have profound applications in various fields, including electrical engineering, quantum mechanics, signal processing, and control theory. They provide a powerful framework for solving problems that cannot be easily addressed using only real numbers. If you're interested in learning more about complex numbers and their fascinating properties, exploring resources on complex analysis can be incredibly rewarding. For a deeper dive into the mathematical concepts and their applications, I recommend checking out Wikipedia's page on Complex Numbers, which offers a comprehensive overview and links to further specialized topics.