Problem 79 - Intermediate Algebra

Question

Simplify. Write the result in the form \(a+b i\) $$ (3+4 i)(2-3 i) $$

Step-by-Step Solution

Answer

The simplified form is \(18 - i\).

Step 1: Apply the distributive property

Start by applying the distributive property (also known as the FOIL method for binomials) to the expression \((3+4i)(2-3i)\). This means you multiply each term in the first parenthesis by each term in the second parenthesis: \((3)(2) + (3)(-3i) + (4i)(2) + (4i)(-3i)\).

Step 2: Multiply the real terms

Calculate the product of the real parts separately: \((3)(2) = 6\).

Step 3: Multiply the real by imaginary terms

Next, multiply the real number by the imaginary part: \((3)(-3i) = -9i\) and \((4i)(2) = 8i\).

Step 4: Multiply the imaginary terms

Now, multiply the imaginary terms: \((4i)(-3i)\). Recall that \(i^2 = -1\), so \((4)(-3)i^2 = -12(-1) = 12\).

Step 5: Combine like terms

Combine all calculated parts: \(6 + 8i - 9i + 12\). Combine the real parts: \(6 + 12 = 18\), and the imaginary parts: \(8i - 9i = -1i\).

Step 6: Write in the form \(a + bi\)

Thus, the expression becomes \(18 - 1i\) or simply \(18 - i\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that states you can multiply a single term by two or more terms inside a set of parentheses. It applies to both real numbers and complex numbers. When dealing with expressions like \((3+4i)(2-3i)\), the distributive property involves multiplying each term in the first parenthesis by every term in the second. This results in four distinct multiplication tasks: \((3)(2) + (3)(-3i) + (4i)(2) + (4i)(-3i)\).
This property is not only a strong foundation for algebraic calculations but also aids in simplifying complex expressions efficiently.

• For real numbers, it ensures that every part of one expression is 'distributed' over the other.
• In the context of complex numbers, it helps in breaking down the multiplication into manageable parts.

Mastering this property enables easier handling of complex equations, simplifying them step by step until they reach their simplest form.

Encountering the Imaginary Unit

The imaginary unit, represented as \(i\), is an essential element of complex numbers. It is defined by the property that \(i^2 = -1\), making it a unique number whose square is negative. Within expressions that involve terms such as \(4i\) and \(-3i\),\(i\) serves as the imaginary component.

• Imaginary numbers are combined with real numbers to form complex numbers that are typically written in the form \(a+bi\), where \(a\) is the real part and \(b\) is the imaginary part.
• In algebraic manipulations, recall that multiplying \(i\) with itself will yield \(-1\), as seen in \((-3i)(4i) = (4)(-3)i^2 = -12(-1) = 12\).

Understanding how to work with \(i\) is crucial for simplifying complex numbers, allowing you to convert expressions into a more straightforward form.

The FOIL Method Unfolded

The FOIL method is a specialized technique used specifically for multiplying binomials. FOIL stands for First, Outside, Inside, Last, which helps in remembering the order of multiplication.

• First: Multiply the first terms in each binomial: \((3)(2)\).
• Outside: Multiply the outer terms: \((3)(-3i)\).
• Inside: Multiply the inner terms: \((4i)(2)\).
• Last: Multiply the last terms: \((4i)(-3i)\).

Using the FOIL method ensures no term is missed, crucial for accurately simplifying expressions. Even when expanding binomials with complex numbers, FOIL proceeds by handling real and imaginary components separately, later combining them. This structured approach keeps your algebraic work systematic and free from errors, resulting in the simplified form \(18 - i\) for the given problem.