Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions of each equation. Do not use a calculator. $$2 x^{3}+4 x=0$$

Step-by-Step Solution

Answer

The complex solutions are \( x = 0, \; x = i\sqrt{2}, \; x = -i\sqrt{2} \).

Step 1: Identify Common Factor

The given equation is \( 2x^3 + 4x = 0 \). First, identify the common factor in both terms. The common factor here is \( 2x \).

Step 2: Factor the Equation

Factor out the common factor from the equation: \( 2x(x^2 + 2) = 0 \). This equation suggests two separate conditions where the product equals zero.

Step 3: Apply Zero Product Property

According to the zero product property, if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \). Apply this property to \( 2x(x^2 + 2) = 0 \). Split it into two equations: \( 2x = 0 \) and \( x^2 + 2 = 0 \).

Step 4: Solve for the First Equation

Solve \( 2x = 0 \): Divide both sides by 2 to isolate \( x \). \( x = 0 \).

Step 5: Solve for the Second Equation

Solve \( x^2 + 2 = 0 \): Subtract 2 from both sides: \( x^2 = -2 \). Take the square root of both sides to find \( x \): \( x = \pm \sqrt{-2} \). Since there is a negative under the square root, express \( x \) in terms of complex numbers: \( x = \pm i\sqrt{2} \).

Step 6: List All Solutions

Combine all solutions from both equations. The solutions are: \( x = 0, \; x = i\sqrt{2}, \; x = -i\sqrt{2} \).

Key Concepts

Zero Product Property

The zero product property is a fundamental concept in algebra. Essentially, it states that if the product of two numbers or expressions is zero, then at least one of the factors must be zero.
For example, if you have an equation like \( ab = 0 \), the zero product property allows you to conclude that either \( a = 0 \) or \( b = 0 \) (or both).
This property is extremely useful when solving polynomial equations because it enables us to break down and solve simpler equations where each factor is set to zero.

• Imagine breaking down a problem into smaller, more manageable parts.
• This approach transforms complex equations into simpler ones.

Applying this property can dramatically simplify finding solutions to polynomial equations, particularly when the polynomials can be factored into linear terms quickly.

Factoring

Factoring is a method used to write a mathematical expression as a product of its factors. In algebra, we often factor expressions to simplify equations or find their solutions.
For instance, consider the equation \( 2x^3 + 4x = 0 \). Notice that both terms are divisible by \( 2x \), which is the greatest common factor.
Thus, the equation can be rewritten as \( 2x(x^2 + 2) = 0 \).
Factoring is a crucial step when employing the zero product property, as it breaks down expressions into their simplest parts, revealing any hidden solutions.

• By identifying common factors, you prepare the equation for easier manipulation.
• This method uncovers simpler expressions to solve individually.

Mastering factoring techniques is indispensable for solving complex algebraic equations effectively.

Complex Numbers

Complex numbers arise naturally when dealing with equations that include the square root of negative numbers. They are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined as \( \sqrt{-1} \).
In our problem, finding \( x^2 = -2 \) leads us to use complex numbers.
To solve \( x^2 + 2 = 0 \), we first set \( x^2 = -2 \), and upon taking the square root, we get \( x = \pm i\sqrt{2} \).

• The imaginary unit \( i \) signifies the presence of complex solutions.
• This allows us to extend our number system to account for square roots of negative numbers.

Understanding complex numbers is essential in fields like engineering, physics, and advanced mathematics, where they model phenomena not explained by real numbers alone.

Square Root of Negative Number

Encountering a square root of a negative number often signals the need for complex solutions. The square root of a negative number is not defined within the realm of real numbers; hence, we introduce the concept of the imaginary unit \( i \).
This opens up a new world of numbers: complex numbers.
The imaginary unit \( i \) is defined as \( \sqrt{-1} \). So, when faced with \( \sqrt{-2} \), you rewrite it using \( i \): \( \pm i\sqrt{2} \).

• This process allows you to solve equations that would otherwise be impossible with only real numbers.
• Transforming negative roots into a solvable format is a key step in simplifying mathematical expressions.

Embracing this concept expands your mathematical toolkit, enabling you to solve and understand a broader class of problems.