Problem 84 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is 20, expressed in standard form as \(20 + 0i\).

Step 1: Identify the Given Expression

We start by identifying the given expression \[ (-2 - 4i)(-2 + 4i) \].This is a product of two conjugate complex numbers.

Step 2: Apply the Difference of Squares Formula

Recognize that the expression follows the pattern of \[ (a - b)(a + b) = a^2 - b^2 \].Here, \(a = -2\) and \(b = 4i\). Use this formula to simplify the expression.

Step 3: Compute \(a^2\) and \(b^2\)

Calculate \(a^2\), where \(a = -2\):\[-2^2 = 4\]. Next, calculate \(b^2\), where \(b = 4i\):\[(4i)^2 = 16i^2 = 16(-1) = -16\].

Step 4: Substitute and Simplify

Substitute \(a^2\) and \(b^2\) back into the formula:\[ a^2 - b^2 = 4 - (-16) = 4 + 16 = 20 \].

Step 5: Express in Standard Form

The result is \(20\), which is a real number. In the standard form of a complex number, it is expressed as \(20 + 0i\), where the imaginary part is zero.

Key Concepts

Difference of Squares

The difference of squares is a useful algebraic identity that helps simplify expressions, especially when you encounter two terms that look like a sum and difference. The formula is: \((a - b)(a + b) = a^2 - b^2\). This pattern can dramatically simplify calculations by reducing what appears to be a complicated multiplication problem into a straightforward subtraction of squares. For example, the expression \((-2 - 4i)(-2 + 4i)\) fits exactly into this formula, as \(a = -2\) and \(b = 4i\). Applying the difference of squares, the product becomes \(a^2 - b^2\), which is a much simpler expression to handle once you've calculated both squares.

Complex Conjugates

Complex conjugates are a pair of complex numbers that have the same real component but opposite imaginary components. If you have a complex number \(a + bi\), its complex conjugate is \(a - bi\). These numbers are helpful when you want to eliminate imaginary components in certain expressions, especially when multiplying conjugates.

• Multiplying a complex number by its conjugate converts it into a real number.
• The product is always the sum of the squares of the real and imaginary parts: \(a^2 + b^2\).

This property is precisely why the calculation of \((-2 - 4i)(-2 + 4i)\) results in a real number. The imaginary parts cancel each other out thanks to the nature of conjugates, leaving us with a pure real number.

Standard Form of a Complex Number

The standard form of a complex number is typically expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. It's important to always present your answer in this format, as it clearly distinguishes between the real and imaginary components. In the given problem, after simplifying \((-2 - 4i)(-2 + 4i)\), we end up with \(20\). Though it does not seem to have an imaginary component, it's crucial to express it as \(20 + 0i\). This shows the problem was solved within the framework of complex numbers and clearly indicates there's no imaginary part. This consistency in format makes it easier to understand and communicate results, especially when dealing with complex numbers.