Question

Multiply. Write the answer in standard form. $$ (3-6 i)^2 $$ 44

Short answer

The answer in standard form is \(-27 - 36i\).

Step-by-step solution

Squaring the complex number

Expanding the square, \((3-6i)^2 = (3-6i)(3-6i)\.

Multiplying the real parts and imaginary parts

Multiply the real parts and the imaginary parts separately, then multiply the combination of real part and imaginary part. That gives us, \(3 * 3 = 9\) (real parts), \(-6i * -6i = 36i^2\) (imaginary parts), \(3 * -6i = -18i\) and \(-6i * 3 = -18i\) (combination of real and imaginary). Now, adding these together, we get: \(9 + 36i^2 - 18i - 18i\).

Simplifying the expression

Knowing \(i^2\) is -1, the equation simplifies to \(9 + 36(-1) - 36i\), which equals \(-27 - 36i\).

Key concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the real number system. It is defined by the equation \(i^2 = -1\). This definition is crucial because it allows us to simplify expressions that otherwise seem complicated.
When we multiply two imaginary units, \(i \times i\), we apply this definition to get \(i^2 = -1\). This property is often used in various calculations involving complex numbers, transforming expressions like \(36i^2\) into \(-36\) and making them simpler to handle.
Understanding the imaginary unit is key to working with complex numbers because it forms the basis of their imaginary part.

Standard Form

Complex numbers are written in standard form as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part, while \(bi\) indicates the imaginary part.
Why is this important? Having a standardized way to write complex numbers ensures clear communication and consistent results when performing operations like addition, subtraction, multiplication, and division. It also ties into plotting these numbers on a complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
Whenever you're asked to solve a problem involving complex numbers, make sure your final result is expressed in this standard form. For example, the result \(-27 - 36i\) from the problem is in standard form, making it easier to interpret and use in further calculations.

Complex Number Multiplication

Multiplying complex numbers involves applying the distributive property—much like you would when multiplying binomials in algebra. Consider two complex numbers: \((a + bi)(c + di)\). Here's how it works:

• Multiply the real parts: \(a \times c\).
• Multiply the outside parts (one real, one imaginary): \(a \times di\).
• Multiply the inside parts (one real, one imaginary): \(bi \times c\).
• Multiply the imaginary parts: \(bi \times di = bdi^2\).

Combine all of these: \((a \times c) + (a \times di) + (bi \times c) + (bd \times i^2)\). Use \(i^2 = -1\) to simplify:
\(ac - bd + (ad + bc)i\).
This process results in another complex number, maintaining the form \(x + yi\), which makes further calculations manageable by keeping everything in standard form.

Complex Number Squaring

Squaring a complex number is a specific case of multiplication. In the problem, we had \((3-6i)^2\), which means multiplying \(3-6i\) by itself.
Let's break it down:

• Start with the original number: \((3-6i)\).
• Set up the product: \((3-6i)(3-6i)\).
• Expand using the method described in complex number multiplication: \((3 \times 3) + (3 \times -6i) + (-6i \times 3) + (-6i \times -6i)\).

Simplify each part:

• Real part: \(3 \times 3 = 9\).
• Imaginary parts: \(-18i - 18i = -36i\).
• Imaginary square: \(-6i \times -6i = 36i^2 = -36\), where \(i^2 = -1\).

Combine these to get the final result in standard form: \(-27 - 36i\). By understanding this process, you'll be better equipped to handle any complex squaring challenge you come across.