Problem 34 - Algebra 2 and Trigonometry

Question

In \(26-37,\) find each product. $$ (5-3 i)(5+3 i) $$

Step-by-Step Solution

Answer

The product (5 - 3i)(5 + 3i) is 34.

Step 1: Identify the Product Form

The problem involves multiplying two binomials: \((5 - 3i)\) and \((5 + 3i)\). This is a difference of squares formula because it takes the form \((a - b)(a + b)\).

Step 2: Apply the Difference of Squares Formula

For any complex numbers, the difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). Here, we identify \(a = 5\) and \(b = 3i\).

Step 3: Square the Real and Imaginary Parts

Calculate \(a^2\) and \(b^2\): \[ a^2 = (5)^2 = 25 \]\[ b^2 = (3i)^2 = 9i^2 = -9 \] Remember that \(i^2 = -1\).

Step 4: Substitute and Simplify

Now substitute these into the difference of squares formula: \[ a^2 - b^2 = 25 - (-9) = 25 + 9 = 34 \]

Step 5: Verify the Solution

The final result of the product \((5 - 3i)(5 + 3i)\) simplifies to the real number \(34\), confirming the calculation is correct.

Key Concepts

Difference of Squares

The concept of the difference of squares is a powerful tool in mathematics that simplifies the multiplication of two binomials in the form \((a - b)(a + b)\). This recognizable pattern results in a simplified expression, \(a^2 - b^2\). This formula works particularly well in cases where one of the binomials has the subtraction operator and the other the addition operator. By observing the expression \((5-3i)(5+3i)\), we identify it as a difference of squares. Here, \(a\) is 5, and \(b\) is \(3i\). The subtraction and addition of the same terms indicate the formula is applicable. This results in:

• \(a^2 = 5^2 = 25\)
• \(b^2 = (3i)^2\) which simplifies to \(-9\) by taking into account that \(i^2 = -1\)
• \(a^2 - b^2 = 25 - (-9) = 34\)

With this simple formula, complex numbers get easily reduced into simpler, more manageable ones. The key is recognizing the pattern and applying the difference of squares formula.

Imaginary Unit

At the heart of complex numbers lies the imaginary unit, denoted by \(i\). This unit is defined with the property that \(i^2 = -1\). Understanding this fundamental unit is crucial for simplifying expressions and solving problems involving complex numbers. The imaginary unit allows us to extend the real number system, thereby enabling the solution of equations that have no real solutions. Consider the expression \((3i)^2\):

• Calculating \((3i)^2\) yields \(9i^2\)
• Substituting \(i^2 = -1\), we find that \(9i^2 = 9(-1) = -9\)

This transformation shows how applying the property of the imaginary unit can transform a seemingly complex problem into something straightforward. Always remember \(i\) is your gateway to operating within the complex number system with elegance.

Binomials

Binomials are expressions containing two terms separated by a plus or minus sign. In complex numbers, these terms often involve the imaginary unit \(i\). Binomials can be multiplied using algebraic methods that are well-known like the distributive property, or simplified by recognizing patterns like the difference of squares.For the binomials \((5-3i)\) and \((5+3i)\) involved in our problem, they hold the pattern of the difference of squares. This pattern makes their product easier to compute. Each binomial consists of a real term and an imaginary term:

• Real term: 5
• Imaginary term: \(-3i\) or \(+3i\)

Binomials are essential for forming the building blocks of more complex algebraic expressions. Recognizing them in problems helps identify potential shortcuts and formulas for faster simplification and solution.

Multiplying Complex Numbers

Multiplying complex numbers involves special rules. When multiplying such numbers, these often include one real and one imaginary component. A common strategy is to use the distributive property, or to simplify using special formulas like the difference of squares when applicable.In our case, with the multiplication \((5-3i)(5+3i)\), you can notice the special pattern. Multiplying these binomials directly involves distributing every term:

• First, multiply real by real: \(5 \times 5 = 25\)
• Then, multiply real by imaginary: \(5 \times 3i = 15i\) and \(-3i \times 5 = -15i\)
• Lastly, multiply imaginary by imaginary: \(-3i \times 3i = -9i^2\)
• Simplify: since \(i^2 = -1\), \(-9i^2 = 9\)
• Combine terms: \(25 + 9 = 34\)

Through these steps, the inherently complex operation of multiplying complex numbers can be simplified down to straightforward arithmetic.