Problem 69 - Algebra for College Students

Question

Find all real number solutions for each equation. $$6 x^{3}+24 x=0$$

Step-by-Step Solution

Answer

The only real solution is x = 0.

Step 1: Identify Common Factor

The first step is to factor out the common factor in the equation. In this case, both terms in the equation 6x^3 + 24x = 0 have a common factor of 6x. Thus, factor out 6x: 6x(x^2 + 4) = 0.

Step 2: Apply Zero Product Property

Next, use the zero product property which states that if the product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero: 6x = 0 or x^2 + 4 = 0.

Step 3: Solve First Factor Equation

Solve the equation 6x = 0. Divide both sides by 6 to isolate x: x = 0.

Step 4: Solve Second Factor Equation

Solve the equation x^2 + 4 = 0. Subtract 4 from both sides to isolate the x^2 term: x^2 = -4. However, this equation has no real solutions because the square of a real number cannot be negative.

Step 5: Compile Real Solutions

Since only the first factor equation has a real solution, the only real solution to the original equation is x = 0.

Key Concepts

Common Factor

In algebra, identifying a common factor is a key step in simplifying and solving equations. A common factor is a number or expression that divides all the terms in an equation without leaving any remainder. Finding the common factor helps in factoring the entire expression, which can simplify the equation into a more easily solvable form.
For the equation \(6x^3 + 24x = 0\), both terms share a common factor of \(6x\). Extracting this common factor means you factor out \(6x\) from each term, leaving you with:

• \(6x(x^2 + 4) = 0\)

This step is crucial as it breaks down the equation into simpler factors, allowing for further analysis and solving. Recognizing and factoring out the common factor is often the first major step in solving polynomial equations.

Zero Product Property

One of the fundamental properties used in solving equations is the zero product property. This property states that if the product of two or more factors is zero, at least one of the factors must be zero. It's a powerful concept because it allows us to set up individual equations for each factor.
In the equation \(6x(x^2 + 4) = 0\), the zero product property lets us write:

• \(6x = 0\)
• \(x^2 + 4 = 0\)

By considering each factor separately, we can solve for the possible values of \(x\). Keep in mind that the property is only valid when the entire expression is set to zero. This step is typically useful for polynomial expressions that have been factored already.

Real Number Solutions

Finding real number solutions involves determining which of the possible solutions from our factored equation are real numbers. In mathematics, real numbers are those that can be found on the number line, including whole numbers, fractions, and decimals, but excluding imaginary numbers.
Solving the factor \(6x = 0\), you simply divide both sides by 6, which gives:

• \(x = 0\)

This is a real solution.
For \(x^2 + 4 = 0\), solving gives us \(x^2 = -4\). However, the square of a real number cannot be negative, implying there are no real number solutions for this equation. Instead, this would have imaginary solutions, but they are outside the scope of real number solutions. Thus, the only real solution to the original equation is \(x = 0\).