Problem 68 - Algebra and Trigonometry

Question

Find all solutions of the equation, and express them in the form \(a+b i\) $$ z+4+\frac{12}{2}=0 $$

Step-by-Step Solution

Answer

The solution is \(-10 + 0i\).

Step 1: Simplify the equation

First, simplify the equation \( z + 4 + \frac{12}{2} = 0 \). Calculate \( \frac{12}{2} \), which equals 6. Substituting back into the equation gives \( z + 4 + 6 = 0 \).

Step 2: Combine constants

Combine the constants 4 and 6 from the equation \( z + 4 + 6 = 0 \). This results in \( z + 10 = 0 \).

Step 3: Solve for z

Isolate \( z \) by subtracting 10 from both sides of the equation: \( z = -10 \).

Step 4: Express as a complex number

Express \( z = -10 \) in the form \( a + bi \). Since there is no imaginary part, \( b = 0 \). Thus, \( z = -10 + 0i \).

Key Concepts

Simplifying Equations

Sometimes, mathematical equations look more complex than they really are, but simplifying them can help us find solutions more easily. Simplifying involves manipulating an equation so it becomes clearer or easier to solve.

In the original exercise, the equation is given as \( z + 4 + \frac{12}{2} = 0 \). The fraction \( \frac{12}{2} \) simplifies directly to 6 because division is just simplifying the relationship between two numbers.

Next, you rewrite the equation by substituting 6 for \( \frac{12}{2} \), leading to \( z + 4 + 6 = 0 \). The constants 4 and 6 can be combined to make the equation easier: combine them to get 10. This results in the simplified form of the equation: \( z + 10 = 0 \).

Simplifying is about performing arithmetic operations step by step and combining like terms whenever possible. This process makes it easier to see the relationships in the equation and focus on finding the solution leading to a neater, simpler form.

Complex Solutions

Complex numbers, noted as \( a + bi \), come into play when dealing with equations involving both real and imaginary parts. In the equation provided, our final simplified form is \( z + 10 = 0 \), and solving this gives \( z = -10 \).

At first glance, you might think this solution is simply a real number. However, expressing it in the form of a complex number involves recognizing the imaginary part. There's no imaginary number part in this equation, which means \( b = 0 \). Thus, \( -10 + 0i \) is still a complex number.

Complex solutions are especially important when encountering different types of polynomial or quadratic equations which could involve imaginary numbers. Recognizing real numbers as complex numbers (with \( b = 0 \)) is a fundamental step in understanding complex solutions. Simplified forms of complex numbers make it easier to perform further operations, like adding, subtracting, multiplying, or dividing them.

Imaginary Numbers

Let’s delve into the concept of imaginary numbers, which are key in forming complex numbers. Imaginary numbers arise from the need to solve equations that involve the square root of negative numbers, something impossible within the realm of real numbers.

The unit of imaginary numbers is \( i \), defined as \( i = \sqrt{-1} \). For any positive real number \( a \), the square root of \(-a\) is \( \sqrt{a} \times i \). Imaginary numbers are integral in the field of complex numbers, where \( a + bi \) forms are standard. Here, \( a \) is the real part, and \( b \) (multiplying \( i \)) is the imaginary part.

In the given exercise, although we end up with \( z = -10 + 0i \), it’s crucial to understand that even though \( b = 0 \) implies no imaginary number contribution, the expression is still in a complex number form. Recognizing and manipulating equations to identify real or imaginary components prepare students for more advanced problems involving imaginary numbers, ensuring the full scope of numbers, both real and imaginary, is well understood.