Problem 77 - Intermediate Algebra

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 is \(-9 + 40i\).

Step 1: Write Down the Formula for Squaring a Binomial

To find \((4 + 5i)^2\), you need to use the formula for squaring a binomial, which is \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = 4\) and \(b = 5i\).

Step 2: Calculate the Square of Each Term

Calculate \(a^2\) and \(b^2\):\[a^2 = (4)^2 = 16\]\[b^2 = (5i)^2 = 25i^2\]Since \(i^2 = -1\), we have:\[b^2 = 25(-1) = -25\]

Step 3: Find the Middle Term

Calculate the middle term using \(2ab\):\[2ab = 2 \times 4 \times 5i = 40i\]

Step 4: Combine All the Terms

Combine all parts together using the binomial expansion formula \(a^2 + 2ab + b^2\):\[16 + 40i - 25\]Combine the real parts \(16\) and \(-25\):\[16 - 25 = -9\]Therefore, the expression simplifies to:\[-9 + 40i\]

Step 5: Express the Result in Standard Form

The number \(-9 + 40i\) is already in the standard form \(a + bi\) of a complex number, where \(a\) is the real part and \(b\) is the imaginary part.

Key Concepts

Binomial Expansion

The binomial expansion is a fundamental algebraic tool used to calculate the expansion of the expression \((a + b)^n\). For squaring a binomial, as in our exercise, we use the formula \((a+b)^2 = a^2 + 2ab + b^2\). This formula helps to expand and simplify binomials easily and has applications in various mathematical problems.

• It breaks down the complexity of expressions into simpler parts by calculating each component: \(a^2\), \(b^2\), and \(2ab\).
• Once calculated, these components are then combined to obtain the expanded form of the expression.

This approach not only simplifies calculations but also provides a useful perspective on the underlying structure of polynomials.

Imaginary Unit

The term \(i\), known as the imaginary unit, is a building block for complex numbers. By definition, \(i\) is the square root of -1, which leads to the fundamental property \(i^2 = -1\). This property is crucial when working with expressions involving complex numbers.

• In calculations, especially those involving squaring terms (like \((5i)^2\), as in our example), it's important to remember that multiplying \(i\) by itself results in -1.
• This allows us to convert complex multiplications into real numbers by multiplying by \(-1\).

The introduction of imaginary units opens up a whole new area of mathematics, enabling the solution of equations that do not have real number solutions.

Standard Form of Complex Numbers

The standard form of a complex number is represented as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. Here, \(a\) refers to the real part and \(b\) to the imaginary part. Expressing results in this form allows for clear communication and understanding of complex numbers.

• It separates the real part from the imaginary part, making it easy to identify each component of the number.
• This form is crucial for performing operations (addition, subtraction, multiplication, and division) on complex numbers.

In our example, \(-9 + 40i\) is the result of simplifying the expression using the binomial expansion with the imaginary unit. Recognizing and using the standard form ensures problems involving complex numbers are solved systematically.