Problem 68 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (4-7 i)^{2} $$

Step-by-Step Solution

Answer

The result is \(-33 + 56i\).

Step 1: Recognize the Formula

Recognize that we need to expand the expression \((4-7i)^2\) using the formula for the square of a binomial: \[(a-b)^2 = a^2 - 2ab + b^2\].Here, \(a = 4\) and \(b = \text{-}7i\).

Step 2: Find \(a^2\)

Calculate \(a^2\) where \(a = 4\):\[4^2 = 16\].

Step 3: Find \(-2ab\)

Calculate \(-2ab\) where \(a = 4\) and \(b = -7i\):\[-2 \times 4 \times (-7i) = 56i\].

Step 4: Find \(b^2\)

Calculate \(b^2\) where \(b = -7i\):\[(-7i)^2 = 49i^2\].Since \(i^2 = -1\), it follows that\[49i^2 = 49 \times (-1) = -49\].

Step 5: Combine Results

Combine the results of the previous steps:\[16 + 56i - 49\].Simplify by combining like terms:\[16 - 49 + 56i = -33 + 56i\].Thus, the final result is in the form \(a + bi\).

Key Concepts

Binomial Expansion

When faced with an expression like \((4-7i)^2\), you are dealing with a binomial expansion involving complex numbers. The binomial expansion formula is pivotal in simplifying expressions of the form \((a-b)^2 = a^2 - 2ab + b^2\). In our case, choose \(a = 4\) and \(b = -7i\). This formula allows you to break down the expansion step by step:

• Calculate \(a^2\) which gives you the square of the first term.
• Use \(-2ab\) to combine both terms while accounting for their interaction.
• Find \(b^2\) for the squared second term.

This structured approach simplifies the process, clearly identifying each part of the expression.

Imaginary Numbers

Imaginary numbers are a core component of complex numbers. They revolve around the imaginary unit \(i\), where \(i = \sqrt{-1}\). When squared, \(i^2\) turns into \(-1\). This property is essential when dealing with powers of \(i\). Consider the term in our problem: \((-7i)^2\). By applying the rule \(i^2 = -1\), it simplifies to \(-49\) since:
\[(-7i)^2 = 49i^2 = 49 imes (-1) = -49\]Imaginary numbers are not "imaginary" in the sense they're fake; they're an essential part of calculations in mathematics, especially in fields like engineering and physics.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have equal real parts and opposite imaginary parts. For example, the complex conjugate of \(4-7i\) would be \(4+7i\).
Although the process of finding \((4-7i)^2\) doesn't directly involve using complex conjugates, understanding them is crucial in complex arithmetic. They are particularly useful when dividing complex numbers or simplifying certain expressions to eliminate the imaginary unit \(i\) from denominators when fractions are involved.
By multiplying a complex number by its conjugate, you can obtain a real number result, which is often needed in various algebraic contexts.

Algebraic Operations

Performing algebraic operations with complex numbers involves understanding how to combine like terms. After finding each component of the expansion \( (4-7i)^2 \), you'll need to:

• Simplify the real terms: Combine \(16\) and \(-49\) to get \(-33\).
• Handle the imaginary term separately: The term \(56i\) remains as is since there are no other imaginary terms to combine it with.

Thus, the expression simplifies to \(-33 + 56i\). This involves basic algebraic skills like distributing, factoring, and simplifying to reach a solution. Remember, the result should always be in the form \(a + bi\) for a complex number.