Problem 54 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{5-2 i}{5+2 i} $$

Step-by-Step Solution

Answer

\(\frac{5-2i}{5+2i} = \frac{21}{29} - \frac{20}{29}i\).

Step 1: Identify the Conjugate

For a given complex number in the denominator, find its conjugate. The conjugate of the complex number \(5 + 2i\) is \(5 - 2i\).

Step 2: Multiply Numerator and Denominator

Multiply both the numerator and the denominator by the conjugate of the denominator. So, multiply \(\frac{5-2i}{5+2i}\) by \(\frac{5-2i}{5-2i}\).

Step 3: Simplify the Numerator

Use the distributive property (FOIL) to simplify the numerator: \[(5 - 2i)(5 - 2i) = 5\cdot5 -5\cdot2i - 2i\cdot5 + (2i)^2 = 25 - 10i - 10i + 4i^2.\]Since \(i^2 = -1\), the simplified form becomes \(25 - 20i - 4 = 21 - 20i\).

Step 4: Simplify the Denominator

Multiply the denominator using the difference of squares formula, \[(5 + 2i)(5 - 2i) = 5^2 - (2i)^2 = 25 - 4(-1) = 25 + 4 = 29.\]

Step 5: Write in Standard Form

Divide the simplified numerator by the simplified denominator to express the result in \(a + bi\) form. This gives \[\frac{21 - 20i}{29} = \frac{21}{29} - \frac{20}{29}i.\]So the result is \(\frac{21}{29} - \frac{20}{29}i\).

Key Concepts

Complex Conjugate

In complex numbers, a conjugate is a special form where you change the sign of the imaginary part. If you have a complex number like \(5 + 2i\), its conjugate will be \(5 - 2i\). This concept is particularly useful in simplifications, especially when dealing with division of complex numbers.

Why do we need the conjugate? When you have a denominator with an imaginary part, it can be challenging to handle. By multiplying the numerator and denominator by the conjugate of the denominator, you eliminate the imaginary part from the denominator. This is because the product of a number and its conjugate is always a real number, not a complex one.

Difference of Squares

The difference of squares formula is a straightforward mathematical identity that lets us handle expressions in a simple way. It states that \(a^2 - b^2 = (a + b)(a - b)\).

In the context of complex numbers, this formula becomes particularly handy. When you multiply a complex number by its conjugate, you can think of it as using the difference of squares technique. For example, with \((5 + 2i)(5 - 2i)\), the process is straightforward:

• First term squared: \(5^2 = 25\)
• Second term squared (consider \(i^2 = -1\)): \((2i)^2 = 4i^2 = -4\)
• Subtract the second from the first: \(25 + 4 = 29\)

This leaves you with a neat real number in the denominator for further simplification.

FOIL Method

The FOIL method is a handy acronym that helps us multiply two binomials, standing for First, Outer, Inner, Last. This method is crucial in simplifying expressions involving complex numbers.

When using FOIL on \((5 - 2i)(5 - 2i)\):

• First: Multiply the first terms: \(5 \times 5 = 25\)
• Outer: Multiply the outer terms: \(5 \times -2i = -10i\)
• Inner: Multiply the inner terms: \(-2i \times 5 = -10i\)
• Last: Multiply the last terms: \(-2i \times -2i = 4i^2\)

Combine these results and remember that \(i^2 = -1\), which transforms \(4i^2\) into \(-4\) in the expression, giving us \(25 - 20i - 4\), which simplifies to \(21 - 20i\).
This method aids in making calculations clear and systematic, ensuring that all parts of the multiplication are accounted for.

Standard Form

When dealing with complex numbers, the goal is often to express the result in standard form, \(a + bi\), where \(a\) and \(b\) are real numbers. This form clearly separates the real and imaginary parts of the number.

Upon simplifying a complex fraction, as in the exercise, we get something like \(\frac{21 - 20i}{29}\). To convert this into standard form, divide each part of the expression by the real number denominator:

• Real part: \(\frac{21}{29}\)
• Imaginary part: \(\frac{20}{29}i\)

Thus, the final expression in standard form becomes \(\frac{21}{29} - \frac{20}{29}i\). By presenting complex numbers in this standard form, it's easier to interpret and perform further calculations with them.