Problem 24 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$\frac{-3-2 i}{5+2 i}$$

Step-by-Step Solution

Answer

\( -\frac{19}{29} - \frac{4}{29} i \)

Step 1: Identify the Need for a Conjugate

To simplify the expression \(\frac{-3-2i}{5+2i}\) in the form \(a + bi\), we need to eliminate the imaginary part from the denominator. To do this, we'll multiply both the numerator and the denominator by the conjugate of the denominator, \(5-2i\).

Step 2: Multiply by the Conjugate

The expression now becomes \[(-3-2i) \times (5-2i)\]\[- (5+2i) \times (5-2i)\].The conjugate is \(5-2i\), and multiplying the denominator this way creates a real number since \[(5+2i)(5-2i) = 5^2 - (2i)^2 = 25 - (-4) = 29\].

Step 3: Expand the Numerator

Expand the numerator \((-3 - 2i)(5 - 2i)\) using distributive property (FOIL Method): \[-3 \times 5 = -15-3 \times -2i = 6i-2i \times 5 = -10i-2i \times -2i = 4i^2.\]Combine these results noting that \(i^2 = -1\): \[-15 + 6i - 10i + 4(-1) = -15 - 4 + (-4i) = -19 - 4i\].

Step 4: Simplify the Expression

Now, write the expression \[\frac{-19-4i}{29}\]in the form \(a + bi\) by separating the terms: \[\frac{-19}{29} + \frac{-4}{29}i\].This gives \(a = \frac{-19}{29}\) and \(b = \frac{-4}{29}\).

Step 5: Present the Final Answer

The expression in the form \(a + bi\) is \[\frac{-19}{29} + \frac{-4}{29} i.\]This form clearly shows the real and imaginary components, \(a = \frac{-19}{29}\) and \(b = \frac{-4}{29}\).

Key Concepts

Conjugate

In complex numbers, the conjugate of a number is formed by changing the sign of its imaginary part. For instance, if we have a complex number in the form of \(a + bi\), its conjugate will be \(a - bi\). This operation is essential when working with complex fractions, particularly to remove the imaginary unit from the denominator.
By multiplying both the numerator and the denominator with the conjugate of the denominator,

• we change the imaginary part in the denominator to zero,
• which helps us write the expression as a real number.

In our example, the denominator was \(5 + 2i\), so its conjugate is \(5 - 2i\). After multiplying, the imaginary components cancel out, simplifying the expression.

Imaginary Unit

The imaginary unit, denoted as \(i\), represents the square root of -1. In mathematics, this unit is fundamental for working with complex numbers. Whenever you see \(i\) in an equation or expression,

• it's a signal that you're dealing with complex numbers,
• and it allows you to handle the square roots of negative values.

When simplifying expressions, remember that \(i^2 = -1\). This property is crucial because when you encounter \(i^2\) during expansion, you can substitute it with -1. This substitution helps in simplifying terms and combining like terms effectively, as seen in our exercise's third step.

FOIL Method

The FOIL method stands for First, Outside, Inside, Last, and is a helpful technique for expanding two binomials in algebra. This acronym guides the multiplication process:

• First: Multiply the first terms of each binomial.
• Outside: Multiply the outer terms in the product.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

In our exercise, expanding the numerator \((-3 - 2i)(5 - 2i)\) requires using the FOIL method. This approach ensures that all components are multiplied correctly, and when combined with simplifying \(i^2 = -1\), leads to the correct expression for the numerator.

Simplifying Fractions

Simplifying complex fractions involves breaking down a complex expression into a simpler form where you can easily identify the real part and the imaginary part. In our exercise, once the numerator and denominator are simplified, we compose the fraction in the form \(a + bi\):

• Separate the real components and the imaginary components.
• Divide both separately by the real number in the denominator.

Hence, from \[ \frac{-19 - 4i}{29} \] we achieve a beautifully expressed result of \[ \frac{-19}{29} + \frac{-4}{29}i \] indicating the specific real and imaginary parts clearly without any combined fraction format.