Problem 77 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(4+5 i)^{2}$$

Step-by-Step Solution

Answer

The product \((4 + 5i)^2\) is \(-9 + 40i\).

Step 1: Apply the formula for squaring a binomial

When squaring any binomial \( (a + b)^2 = a^2 + 2ab + b^2 \). Here, let \( a = 4 \) and \( b = 5i \). We will apply this formula.

Step 2: Calculate \(a^2\)

Calculate the square of the real part, \( a^2 = 4^2 = 16 \).

Step 3: Calculate \(2ab\)

Calculate the double product of the real and imaginary parts, \( 2ab = 2 \times 4 \times 5i = 40i \).

Step 4: Calculate \(b^2\)

Calculate the square of the imaginary part, \( b^2 = (5i)^2 = 25i^2 \). Recall that \( i^2 = -1 \), so we have \( 25i^2 = 25(-1) = -25 \).

Step 5: Add the results to express in standard form

Combine the results from steps 2, 3, and 4: \( a^2 + 2ab + b^2 = 16 + 40i - 25 = -9 + 40i \). The standard form of a complex number is \( a + bi \), so the expression is \(-9 + 40i\).

Key Concepts

Binomials

In the context of algebra, a binomial is an expression that contains two distinct terms. The term comes from the Latin prefix 'bi-', meaning two, and it's often expressed in the form \( a + b \) or \( a - b \). The components of a binomial can include constants, variables, or both. Binomials are important because they frequently appear in algebraic equations and are fundamental in operations such as expansion, factoring, and even calculus. When dealing with binomials, squaring is a common operation. To square a binomial like \((a + b)^2\), we apply the formula:

• \( a^2 + 2ab + b^2 \)

This formula is derived from the distributive property of multiplication over addition, where you multiply each term in the first binomial by each term in the second. Lets say the binomial is \( (4 + 5i) \), the process involves calculating each component separately, such as \( a^2 \), \( 2ab \), and \( b^2 \). This can provide a simpler way to manage complex algebraic manipulations.

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics that extends our understanding beyond real numbers. An imaginary number is essentially a multiple of \( i \), where \( i \) is the imaginary unit defined as \( \sqrt{-1} \). This means \( i^2 \) equals \(-1\).Imaginary numbers are often treated as 'mysterious' because they arise from taking the square root of a negative number—something not possible in the realm of real numbers. However, they become incredibly useful in various fields such as engineering, physics, and most importantly, complex number theory.

• \( i \) is the basis of all imaginary numbers
• Any number that can be expressed in the form \( bi \) (where \( b \) is a real number) is an imaginary number.

For example, in the expression \( 5i \), the number 5 is the real number coefficient, and when squared, it illustrates how \( (5i)^2 = 25i^2 = -25 \). Imaginary numbers thus work in concert with real numbers to form complex numbers.

Standard Form of Complex Numbers

Complex numbers are an extension of real numbers and can be expressed in standard form as \( a + bi \). Here, \( 'a' \) represents the real part, while \( 'b' \) represents the imaginary part, along with the imaginary unit \( i \). Complex numbers allow us to capture a full spectrum of solutions, especially in quadratic equations, where roots might otherwise be left unaccounted for.The standard form is essential for simplifying calculations and for interpreting the properties of complex numbers clearly. When complex numbers are squared, or undergo other algebraic manipulations, their results are typically simplified back into this form.For instance, consider the expression \( 4+5i \). When squared using the binomial expansion, you find:

• Real part: \( 4^2 = 16 \)
• Imaginary part: \( 2 \times 4 \times 5i = 40i \)
• Second imaginary square: \( (5i)^2 = 25i^2 = -25 \)

Add these to get the final expression: \( 16 + 40i - 25 = -9 + 40i \), which is in the standard form of a complex number. Understanding and using this form is pivotal in managing calculations involving complex solutions effectively in mathematics.