Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Multiply as indicated. Write each product in standand form. $$(-3+2 i)^{2}$$

Step-by-Step Solution

Answer

The standard form of the product is \(5 - 12i\).

Step 1: Understand the Problem

We need to square the complex number \((-3 + 2i)\). This means multiplying \((-3 + 2i)\) by itself.

Step 2: Apply the Formula for Squaring a Binomial

The square of a binomial \((a + b)^2\) can be expanded using the formula: \((a + b)^2 = a^2 + 2ab + b^2\). For \((-3 + 2i)\), let \(a = -3\) and \(b = 2i\).

Step 3: Calculate Each Part of the Expression

First, calculate \[a^2 = (-3)^2 = 9\]Then, calculate \[2ab = 2(-3)(2i) = -12i\]And finally, calculate \[b^2 = (2i)^2 = 4i^2\]. Recall that \(i^2 = -1\), so \[4i^2 = 4(-1) = -4\]

Step 4: Combine the Results

Add the results from Step 3: \[9 + (-12i) + (-4) = 9 - 12i - 4\]Combine the real parts and the imaginary part:\[9 - 4 = 5\]Thus, we get \[5 - 12i\] as the final expression.

Key Concepts

Binomial Expansion

Binomial expansion is a valuable algebraic technique. It's used to expand expressions that are raised to a power, like \((a + b)^n\). For a binomial powers like \((a + b)^2\), the expansion formula is \((a^2 + 2ab + b^2)\). We apply this to split the original expression into simpler pieces.
In our example, we expand \((-3 + 2i)^2\). Here:

• \(a\) is \(-3\)
• \(b\) is \(+2i\)

Applying the formula step-by-step involves:

• Calculating \((a^2)\):\((-3)^2 = 9\)
• Calculating \(2ab\): \(2(-3)(2i) = -12i\)
• Finally, \((b^2)\): \( (2i)^2 = 4i^2\)

This expansion helps simplify the process of multiplying complex numbers like \((-3+2i)^2\). It breaks down operations and makes them easier to compute further on.

Imaginary Unit

The imaginary unit, represented as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This intriguing definition allows us to perform arithmetic with the so-called 'imaginary' numbers.
When dealing with expressions involving \(i\), such as \((2i)^2\), we must remember this key property. In our step-by-step solution, we saw how when \( (2i)^2 = 4i^2 \) it simplifies to \(-4\), using \(i^2 = -1\).

• When squaring \(i\), replace \(i^2\) with \(-1\)
• Multiply \(i^2\) by any scalar square, in our case \((4)\)

This gives us practical tools to navigate calculations involving \(i\), converting potential complexity into conventional real arithmetic.

Standard Form of Complex Numbers

Complex numbers are often expressed in their standard form, written as \((a + bi)\), where \(a\) is the real part and \(bi\) is the imaginary part.
After computation using expansion or other methods, it's important to express results in standard form to give a clear representation of both the real and imaginary components.
In our example throughout this problem, after calculating \(-3 + 2i)^2\), we arrived at \((5 - 12i)\).

• The real part here is \(5\)
• The imaginary part is \(-12i\)

This format allows quick identification of both parts of a complex number, ensuring clarity and ease of further mathematical processes.