Problem 12 - College Algebra with Modeling and Visualization

Question

Simplify by using the imaginary unit \(i\). $$ \frac{-2 \pm \sqrt{-36}}{6} $$

Step-by-Step Solution

Answer

The simplified form is \(-\frac{1}{3} \pm i\).

Step 1: Identify the Imaginary Component

The expression under the square root, \(-36\), is negative. Therefore, we will need to use the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-36} = \sqrt{36} \times \sqrt{-1} = 6i\).

Step 2: Substitute the Imaginary Component

Substitute \(6i\) into the original equation in place of \(\sqrt{-36}\). This transforms the equation to \(\frac{-2 \pm 6i}{6}\).

Step 3: Simplify the Expression by Splitting the Fraction

The expression \(\frac{-2 \pm 6i}{6}\) can be split into two separate fractions: \(\frac{-2}{6} \pm \frac{6i}{6}\).

Step 4: Simplify Each Fraction Separately

Simplify each part of the fraction individually: \(\frac{-2}{6} = -\frac{1}{3}\) and \(\frac{6i}{6} = i\).

Step 5: Combine Simplified Components

Combine the simplified fractions into the expression: \(-\frac{1}{3} \pm i\).

Key Concepts

Complex Numbers

Complex numbers are an essential part of mathematics, especially when dealing with equations involving square roots of negative numbers. A complex number consists of a real part and an imaginary part. This can be expressed in the form: \[ a + bi \] where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit is denoted by \( i \), which is defined by the property \( i^2 = -1 \).

• Example: If \( a = 2 \) and \( b = 3 \), then the complex number is \( 2 + 3i \).
• This means 2 is the real part and 3i is the imaginary part.

Complex numbers allow mathematicians to perform calculations that take into account both the real and imaginary components, which is crucial for solving certain algebraic equations.

Simplification

Simplification is the process of reducing a complex mathematical expression to its simplest form. This can be done by performing basic arithmetic operations and combining like terms. In our exercise, we simplified the expression \( \frac{-2 \pm 6i}{6} \) by dividing both the real and imaginary parts by 6. This means computing:

• \( \frac{-2}{6} = -\frac{1}{3} \)
• \( \frac{6i}{6} = i \)

By simplifying the equation, we achieve a concise representation: \(-\frac{1}{3} \pm i \). This step is important because it makes the expression easier to understand and work with, providing a clearer picture of the solution. It also reflects the "real number plus imaginary number" format typical of complex numbers.

Square Roots

Square roots are values that, when multiplied by themselves, yield the original number. However, when dealing with negative numbers under the square root, we utilize the concept of imaginary numbers. The negative square root part, such as \( \sqrt{-36} \), involves a twist because the principal square root of a positive number and the imaginary unit \( i \) needs to be considered.
The procedure goes as follows:

• Calculate the square root of the positive part: \( \sqrt{36} = 6 \).
• Incorporate the imaginary unit: \( \sqrt{-36} = 6i \).

Using \( i \) allows us to work with square roots of negative numbers, leading to a broader range of solutions in algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Simplifying them is a key skill in algebra that involves recognizing and manipulating these components to find solutions. In our given problem, the expression is modified through several transformations like the introduction of the imaginary number \( i \) and the simplification of fractions. Here's how they are structured:

• Original expression: \( \frac{-2 \pm \sqrt{-36}}{6} \)
• Using imaginary numbers: substitute \( 6i \) for \( \sqrt{-36} \)
• Simplify after substitution: \( \frac{-2 \pm 6i}{6} \)
• Simplified form: \(-\frac{1}{3} \pm i \)

Understanding how to manipulate these expressions, including both the real and imaginary parts, is vital in solving and simplifying algebraic problems. It ensures that one can express complex solutions in manageable and well-defined terms.