Problem 79 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (-2-4 i)^{2} $$

Step-by-Step Solution

Answer

The result is \(-12 + 16i\), in standard complex form.

Step 1: Recognize the Complex Number Form

The given expression has the form \( (-2-4i) \), where \( a = -2 \) and \( b = -4 \). This is in the form of a complex number \( a + bi \).

Step 2: Apply the Square Formula

Use the formula for squaring a binomial \((a + bi)^2 = a^2 + 2abi + (bi)^2\). Substitute \( a = -2 \) and \( b = -4 \) into the formula.

Step 3: Calculate Each Term Separately

Calculate \( (-2)^2 = 4 \), \( 2 imes (-2) imes (-4i) = 16i \), and \( (-4i)^2 = 16i^2 \).

Step 4: Simplify Using \( i^2 = -1 \)

Since \( i^2 = -1 \), replace \( i^2 \) with \(-1\) in the expression: \( 16i^2 = 16(-1) = -16 \).

Step 5: Combine Real and Imaginary Parts

Combine the calculated terms: \( 4 \) (from \( (-2)^2 \)), \(-16 \) (from \( (-4i)^2 \)), and \( 16i \) (from the middle term). This gives \( 4 - 16 + 16i \).

Step 6: Express in Standard Form

Combine the real numbers \( 4 \) and \(-16 \) to get \(-12\) and keep the imaginary part \( 16i \). This results in the standard form \(-12 + 16i \).

Key Concepts

Standard Form

When we talk about the standard form of a complex number, we are looking at the expression in the form of \( a + bi \). Here, \( a \) is the real part, while \( b \) represents the imaginary part and \( i \) is the imaginary unit. The standard form helps to separate the real and imaginary parts, making it easier to perform operations with complex numbers.

For example, the complex number \(-12 + 16i\) is in standard form, where \(-12\) is the real part and \(16i\) is the imaginary part. It's essential to express complex numbers in this format to clearly distinguish between the real and imaginary components, which is crucial for mathematical operations and interpretations.

Binomial Squaring

Squaring a binomial might seem daunting, but it’s straightforward once you know the formula: \( (a + b)^2 = a^2 + 2ab + b^2 \). When it comes to complex numbers, like in this exercise, the binomial will have an imaginary component.

For example, if you have a complex number like \( (-2 - 4i) \), you can square it using the adapted formula for complex binomials: \( (a + bi)^2 = a^2 + 2abi + (bi)^2 \).

• First, calculate \( a^2 \), which is \((-2)^2 = 4\).
• Next, find \( 2ab \), where you multiply two times the real part by the imaginary part, which is \( 2 imes (-2) imes (-4i) = 16i \).
• Finally, compute \( (bi)^2 \), which is \((-4i)^2 = 16i^2\), and substitute \(i^2\) with \(-1\) to get \(-16\).

Understanding this formula is crucial because it is frequently used in algebra and further studies of complex numbers.

Complex Conjugate

A complex conjugate flips the sign of the imaginary part of a complex number. If you have a complex number \( a + bi \), its complex conjugate will be \( a - bi \). Using complex conjugates can simplify complex number division and multiplication.

In calculations, especially when working with the magnitude of complex numbers or rationalizing denominators, employing the complex conjugate formula is beneficial. Though it wasn't crucial in the direct calculation of \((-2-4i)^{2}\), understanding how conjugates work is vital for a well-rounded understanding of complex numbers.

Imaginary Unit

The imaginary unit, denoted as \( i \), is the cornerstone of complex numbers and is defined as \( i^2 = -1 \). This definition allows us to extend real numbers into the complex plane and introduces a whole new dimension to calculations.

In practical problem-solving, like in our exercise, it plays a critical role when squaring binomials that include complex numbers. Knowing that \( i^2 = -1 \) allows you to substitute and simplify expressions involving \( i \). For instance, in the original exercise, recognizing this relationship helps transform a term like \( 16i^2 \) into \(-16\).

• This transformation gives the real number clarity and reduces confusion during calculations.
• Understanding the imaginary unit is foundational to tackling more complex calculus and engineering problems.

Remembering that \( i \) opens up the mathematical possibilities beyond the visible real number line into the imaginary world is crucial for any math student venturing into complex numbers.