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Chapter 7: Problem 64

Multiply and simplify. Write each answer in the form \(a+b i\). $$(-5-2 i)^{2}$$

Short Answer

Expert verified
21 + 20i

Step by step solution

01

Understand the Problem

The task is to multiply and simplify the expression \((-5 - 2i)^{2}\), and then write the answer in the form \(a + bi\).
02

Apply the Binomial Theorem

Recall that \( (a - b)^{2} = a^{2} - 2ab + b^{2} \). Here, \(a = -5\) and \(b = 2i\).
03

Substitute the Values

Substitute the values into the formula: \[ (-5 - 2i)^{2} = (-5)^{2} - 2 \times (-5) \times (2i) + (2i)^{2} \]
04

Calculate Each Term

Compute each term separately: \((-5)^{2} = 25\), \(-2 \times (-5) \times 2i = 20i\), and \( (2i)^{2} = 4i^{2} = 4(-1) = -4 \).
05

Combine the Terms

Combine the results of each term to get: \[ 25 + 20i - 4 \]
06

Simplify the Expression

Combine the real parts and imaginary parts: \[ 25 - 4 + 20i = 21 + 20i \]

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Key Concepts

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

Binomial Expansion
To solve problems like multiplying \( (-5 - 2i)^2 \) e use the binomial expansion. When we expand terms like \( (a - b)^2 \) we get the formula \( a^2 - 2ab + b^2 \). In our context, \( a = -5 \) and \( b = 2i \).
听
This binomial expansion splits the multiplication into smaller, simpler parts. We compute three terms: \( (-5)^2, -2 \times (-5) \times 2i, (2i)^2 \).
听
Remember:
  • Compute each part separately.
  • Use the binomial formula for easier calculation.
  • Take care of the signs and coefficients, especially for complex parts.
Imaginary Unit
In complex numbers, \( i \) is called the imaginary unit. \( i \) is defined as \( \sqrt{-1} . \) This means:
  • When we see \( i^2, \) it equals \( -1. \)
  • Imaginary numbers typically have the form \( bi \), where \ (b) is a real number.
For example, in our step-by-step solution when we calculate \( (2i)^2, \) we turn it into \( 4i^2 = 4(-1) = -4. \)

This simplification helps in combining real and imaginary terms.
Knowing how to handle the imaginary unit is crucial for working effectively with complex numbers.
Combining Like Terms
After breaking down and computing all parts of the binomial expansion, we combine like terms. Real numbers combine with real numbers, and imaginary parts combine with imaginary parts.
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Example: In our problem, the terms break down to \( 25, 20i, \) and \( -4.\)
Combine these:
  • Combine real numbers: \( 25 \) and \( -4 \) to get \( 21. \)
  • Imaginary term remains \( 20i. \)

Result: \ (21 + 20i ) \
. Remember to:
  • Always simplify fully by combining all like terms into a single real and single imaginary part.
  • Write the final answer in the form \( a + bi, \) where both parts are combined and simplified characters.

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