Problem 63 - Intermediate Algebra

Question

Multiply. Write all answers in the form a \(+b i.\) $$ (2+i)^{2} $$

Step-by-Step Solution

Answer

The answer is \(3 + 4i\).

Step 1: Apply the formula for squaring a binomial

The expression you need to expand is \((2 + i)^2\). When squaring a binomial, you use the formula \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2\) and \(b = i\).

Step 2: Square the first term

Calculate \(a^2\), which in this case is \(2^2 = 4\).

Step 3: Square the imaginary unit

Calculate \(b^2\), which is \(i^2 = -1\). Recall that \(i\) is the imaginary unit and \(i^2\) is defined to be \(-1\).

Step 4: Multiply twice the product of the two terms

Calculate \(2ab\), which is \(2 \times 2 \times i = 4i\).

Step 5: Combine the results

Combine the results from steps 2, 3, and 4, using the formula: \(a^2 + 2ab + b^2 = 4 + 4i - 1\).

Step 6: Simplify the expression

Simplify the expression by combining like terms: \(4 - 1 + 4i = 3 + 4i\).

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are powers of a binomial. A binomial is an algebraic expression containing two terms, such as \((a + b)^n\). In this case, expansion helps us break down expressions like \((2+i)^2\) into simpler terms. This makes calculations manageable.
For squaring a binomial, we apply a specific formula: \((a + b)^2 = a^2 + 2ab + b^2\). Here, each part of the formula has a distinct role:

• \(a^2\) represents the square of the first term.
• \(2ab\) is twice the product of both terms.
• \(b^2\) is the square of the second term.

In our given expression \((2+i)^2\), applying the formula involves substituting \(a = 2\) and \(b = i\). Expanding the expression helps convert it into a linear combination of real and imaginary numbers, simplifying further calculations.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept when dealing with complex numbers. It is defined by the property that \(i^2 = -1\). This might seem unusual at first, because we don't encounter such properties with real numbers.
Imaginary numbers extend the number system beyond real numbers by introducing these properties. With \(i\), we can express numbers that aren’t on the real number line, opening up new solutions for equations such as \(x^2 + 1 = 0\).
When working with complex numbers like \(2 + i\), understanding the imaginary unit is crucial. For instance, when you square \(i\) during binomial expansion, knowing that \(i^2 = -1\) changes the imaginary component into a real component, influencing the overall result.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions help represent general mathematical ideas and relationships. When dealing with complex numbers like \(2+i\), algebraic expressions become vital.
They are used throughout the exercises to combine real and imaginary parts into one formulaic expression. For instance, after expanding \((2+i)^2\) using binomial rules, we have components like \(2^2\), \(4i\), and \(i^2\).
Bringing these pieces together leads to the expression \(4 + 4i -1\). This expression then simplifies to \(3 + 4i\) which is crucial for expressing complex numbers in the form \(a + bi\). These principles allow careful manipulation of numbers to simplify, solve and understand relationships between mathematical elements.