Problem 50 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ (6+2 i)(6-2 i) $$

Step-by-Step Solution

Answer

The product is 40.

Step 1: Identify the terms in the expression

The expression to multiply is \((6 + 2i)(6 - 2i)\). Here, we have two complex numbers in the expression.

Step 2: Apply the formula for a difference of squares

When we multiply expressions of the form \((a+bi)(a-bi)\),we use the formula: \((a^2 - (bi)^2)\). This is useful since it helps simplify multiplication without expanding.

Step 3: Calculate the squares for each term

Here, both \(a\) and \(bi\) are identified: \(a = 6\) and \(bi = 2i\). Calculate the squares: \(a^2 = 6^2 = 36\) and \((bi)^2 = (2i)^2 = 4i^2 = 4(-1) = -4\).

Step 4: Substitute the squared terms into the formulation

Substitute the calculated squares back into the formula:\(a^2 - (bi)^2 \= 36 - (-4)\).

Step 5: Simplify the expression

Perform the subtraction:\(36 - (-4) = 36 + 4 = 40\).

Step 6: Write the result in standard form

The final expression in the standard form, where there is no imaginary component, simplifies to \(40 + 0i\).

Key Concepts

Multiplication of Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, often expressed in the form \(a + bi\). Multiplying complex numbers follows a process similar to multiplying binomials. The process involves applying the distributive property or the FOIL method (First, Outside, Inside, Last). This involves multiplying each term in the first complex number by each term in the second complex number, and then combining like terms.

For example, consider multiplying \((6 + 2i)\) and \((6 - 2i)\):

• First: \(6 \times 6 = 36\)
• Outside: \(6 \times (-2i) = -12i\)
• Inside: \(2i \times 6 = 12i\)
• Last: \(2i \times (-2i) = -4i^2\)

The key step is to understand how to handle \(i^2\) in the multiplication, which transitions into a concept known as the imaginary unit.

Difference of Squares Formula

The expression \((6 + 2i)(6 - 2i)\) is structured in a special way called the difference of squares. The difference of squares formula is a key algebraic identity: \((a + b)(a - b) = a^2 - b^2\). This formula simplifies the multiplication process significantly because only two squares are required rather than distributing all terms.

Applying this to complex numbers, we get:

• \((a + bi)(a - bi) = a^2 - (bi)^2\)
• Substitute: \(a = 6\) and \(b = 2i\)
• Simplifies to: \(36 - 4i^2\)

This simplification particularly shows its strength by reducing computational complexity when working with conjugates of complex numbers.

Imaginary Unit i

The imaginary unit \(i\) is foundational in complex numbers. It is defined by the property that \(i^2 = -1\). This property allows us to incorporate negative square roots in calculations, a limitation of real numbers alone. When performing operations like multiplication where \(i\) appears, replacing \(i^2\) with \(-1\) is crucial to simplifying the expression.

In our multiplication problem, the term \((2i)^2\) at first yields \(4i^2\). Here, we substitute \(i^2 = -1\), converting \(4i^2\) into \(4(-1) = -4\). This transformation ensures that the final result is a real number and not a complex number.

Understanding \(i^2 = -1\) is essential to efficiently work with complex numbers, as it often appears during calculations.