Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{2+9 i}{-3-i} $$

Step-by-Step Solution

Answer

The expression is \( -1.5 - 2.5i \).

Step 1: Multiply by the Complex Conjugate

To simplify the division of the complex numbers, multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(-3-i\) is \(-3+i\). So the expression becomes:\[\frac{(2+9i)(-3+i)}{(-3-i)(-3+i)}.\]

Step 2: Simplify the Denominator

The denominator \((-3-i)(-3+i)\) is a difference of squares and simplifies to:\[(-3)^2 - (i)^2 = 9 - (-1) = 9 + 1 = 10.\]

Step 3: Expand the Numerator

Expand the expression \((2+9i)(-3+i)\) using distributive property (FOIL method):\[2(-3) + 2(i) + 9i(-3) + 9i^2.\]Calculate each term:- First: \(2 \times -3 = -6\)- Outer: \(2 \times i = 2i\)- Inner: \(9i \times -3 = -27i\)- Last: \(9i^2 = 9(-1) = -9\)Combine these, yielding:\(-6 + 2i - 27i - 9 = -15 - 25i\).

Step 4: Write the Expression in the Form of a+bi

Now, substitute the simplified numerator and denominator into the expression:\[\frac{-15 - 25i}{10} = \frac{-15}{10} + \frac{-25i}{10}.\]Simplify to obtain:\[-1.5 - 2.5i.\]

Step 5: Final Expression

Thus, \(\frac{2+9i}{-3-i}\) in the form \(a+bi\) is:\[-1.5 - 2.5i.\]

Key Concepts

Complex Conjugate

To understand complex numbers, it's essential to learn about the complex conjugate. When you have a complex number, say, in the form of \(a + bi\), its complex conjugate is \(a - bi\). The central idea is that the imaginary part's sign changes. For instance, if the complex number is \(-3 - i\), its conjugate would be \(-3 + i\).

• Using complex conjugates is crucial when dealing with division in complex numbers. By multiplying both the numerator and the denominator by the conjugate of the denominator, you "remove" the imaginary part from the denominator.
• This process is quite similar to rationalizing a fraction by eliminating radicals.

This mathematical maneuver makes calculations simpler and results easier to interpret in the standard form \(a + bi\).

Difference of Squares

The difference of squares is a valuable algebraic identity. It states that for any two terms \(x\) and \(y\), the identity \((x - y)(x + y) = x^2 - y^2\) holds true. This concept is especially useful when simplifying expressions involving complex conjugates.

• In our exercise, the denominator \((-3-i)(-3+i)\) presents a clear example of the difference of squares.
• By applying the identity, we simplify it as \((-3)^2 - (i)^2\), which further simplifies to \(9 - (-1) = 10\).

Recognizing and applying this identity helps reduce complex calculations to simpler forms, streamlining your work with complex numbers.

Distributive Property

The distributive property, also known as the distributive law of multiplication, states that \(a(b+c) = ab + ac\). In dealing with complex numbers, we often use this property to expand expressions, particularly in multiplication.

• In our example, to simplify the numerator \((2+9i)(-3+i)\), we use the distributive property.
• This expansion follows the FOIL method (First, Outer, Inner, Last), resulting in: \(-6 + 2i - 27i - 9\).
• Simplifying these terms gives \(-15 - 25i\).

Understanding the distributive property aids in breaking down complex number expressions into parts that are easier to manage and combine.

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i\) is defined such that \(i^2 = -1\). They fill a crucial role in the field of complex numbers.

• An imaginary number like \(bi\) becomes a component of a complex number \(a + bi\).
• Imaginary numbers help us work with square roots of negative numbers, which don't have solutions in the set of real numbers alone.
• In our exercise, we see \(9i\) and \(-i\) combining in operations that involve multiplication and addition, showing how imaginary components interact with real figures.

Harnessing the logical framework of imaginary numbers allows mathematicians and engineers to tackle problems that go beyond the limit of real-number solutions, offering powerful tools for a wide variety of applications.