Problem 36 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (2-3 i)^{2} $$

Step-by-Step Solution

Answer

The standard form is \(-5 - 12i.\)

Step 1: Expand the expression

The expression \[(2 - 3i)^2\] can be expanded using the formula for squaring a binomial, \((a - b)^2 = a^2 - 2ab + b^2.\) Substituting \(a = 2\) and \(b = 3i\), we get: \[ (2)^2 - 2 \cdot (2) \cdot (3i) + (3i)^2.\]

Step 2: Simplify term-by-term

Now, perform calculations for each of the three terms separately. 1. Calculate the square of the real part: \(2^2 = 4.\)2. Calculate the product of twice the real part and the imaginary part: \( -2 \times 2 \times 3i = -12i.\)3. Calculate the square of the imaginary part: \((3i)^2 = -9.\) (Remember that \(i^2 = -1\), hence \(3i \times 3i = 9i^2 = 9(-1) = -9.\))

Step 3: Combine simplified terms

Combine the results from the previous step:\[4 - 12i - 9.\]Simplify further by combining like terms to write the expression in standard form:\[4 - 9 = -5,\]to get:\[-5 - 12i.\]

Step 4: Write in standard form

The complex expression \((2 - 3i)^2\) is now written in standard form as \(-5 - 12i,\) where \(-5\) is the real part and \(-12i\) is the imaginary part.

Key Concepts

Standard Form of Complex Numbers

Complex numbers are written in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part. This is known as the "standard form" of a complex number.
The real part \( a \) and the imaginary part \( bi \) can be any real numbers. It's important to realize that the imaginary part includes both the coefficient \( b \) and the imaginary unit \( i \).
Standard form is crucial for performing arithmetic operations such as addition, multiplication, and for making comparisons between complex numbers.
In our exercise, the expression \((2-3i)^2\) is eventually simplified to \(-5 - 12i\). Here, \(-5\) is the real part, and \(-12i\) is the imaginary part, conforming to the standard form \( a + bi \). By organizing our result in this way, we make it easier to understand and use in further mathematical operations.

Imaginary Unit

The imaginary unit is a mathematical concept denoted by \( i \) which is defined as the square root of \(-1\). So, \( i^2 = -1 \).
This concept is crucial as it allows us to extend the real number system to include solutions to equations like \( x^2 + 1 = 0 \), which have no real solutions.
The imaginary unit enables us to work with complex numbers, which have both real and imaginary components. When performing calculations, remember:

• Multiplication involving \( i \) should carefully consider \( i^2 = -1 \).
• Squaring an imaginary number, like \( (3i)^2 \), requires using \( i^2 = -1 \), leading to \(-9\) in our example.

Understanding \( i \) helps to simplify expressions and obtain the standard form of complex numbers. In the exercise, \((3i)^2 = -9\) arises directly from applying this property.

Binomial Expansion

The binomial expansion is a method to expand expressions that are raised to a power, like \((a+b)^n\). One common case is when \( n = 2 \), which reads as squaring a binomial.
The general formula for squaring a binomial is \((a + b)^2 = a^2 + 2ab + b^2 \). For \((a - b)^2\), it becomes \(a^2 - 2ab + b^2\), accounting for the sign.
Applying this to our original problem of \((2 - 3i)^2\):

• First Term: Square the first part: \( 2^2 = 4 \).
• Second Term: Twice the product of the first and second parts: \(-2 \times 2 \times 3i = -12i \).
• Third Term: Square the second part: \((3i)^2 = -9 \). Remember, squaring \( 3i \) gives \( 9i^2 \), and since \( i^2 = -1 \), it becomes \(-9 \).

Using binomial expansion simplifies the process of handling complex expressions by breaking them into manageable parts. Ultimately, combining these terms gives the result \(-5 - 12i\).