Problem 15 - Brief Applied Calculus

Question

Solve each equation by factoring. [Hint for Exer cises 19-22: First factor out a fractional power.] $$ 2 x^{3}+18 x=12 x^{2} $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 3\).

Step 1: Move All Terms to One Side

First, move all terms to one side of the equation so the equation equals zero. Start with the original equation: \[ 2x^{3} + 18x = 12x^{2} \] Subtract \(12x^{2}\) from both sides:\[ 2x^{3} + 18x - 12x^{2} = 0 \]

Step 2: Rearrange and Factor Out the Greatest Common Factor

Rearrange the terms in descending order of their exponent:\[ 2x^{3} - 12x^{2} + 18x = 0 \]Next, factor out the greatest common factor, which is \(2x\):\[ 2x(x^{2} - 6x + 9) = 0 \]

Step 3: Factor the Quadratic Expression

Now, factor the quadratic expression inside the parentheses, \(x^{2} - 6x + 9\). Notice it can be factored as a perfect square:\[ (x - 3)^{2} \]Thus, the expression becomes:\[ 2x(x - 3)^{2} = 0 \]

Step 4: Solve for x

Set each factor equal to zero and solve:1. \(2x = 0\), so \(x = 0\).2. \((x - 3)^{2} = 0\). Solving gives \(x - 3 = 0\), so \(x = 3\). Therefore, the solutions are \(x = 0\) and \(x = 3\).

Key Concepts

Quadratic Expressions

A quadratic expression is a type of polynomial that is of the second degree. It generally takes the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratics are crucial in factoring as many equations you encounter will be quadratic or can be manipulated into one.
Understanding the structure of quadratics helps in identifying patterns and methods for factoring them into simpler components.
In the context of the exercise, the quadratic expression inside the parentheses, \( x^2 - 6x + 9 \), is a great example. It is a perfect square trinomial, which means it can neatly be factored into \( (x - 3)^2 \).
When factoring quadratic expressions, you should look for:

• Perfect square trinomials like \( a^2 - 2ab + b^2 = (a-b)^2 \)
• Using the quadratic formula if it doesn't factor easily
• Factoring by grouping, which involves splitting the middle term

Quadratic expressions are the building blocks of many algebraic problems, so mastering them is essential.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides all terms in a polynomial without leaving a remainder. Identifying the GCF is an essential first step in simplifying or factoring any polynomial equation.
In this exercise, once we moved all the terms to one side of the equation, we reordered them (\( 2x^3 - 12x^2 + 18x = 0 \)).
Here, the GCF is \( 2x \) because 2 and \( x \) are common in every term:

• \( 2x \cdot x^2 = 2x^3 \)
• \( 2x \cdot (-6x) = -12x^2 \)
• \( 2x \cdot 9 = 18x \)

By factoring out the GCF, we make the polynomial easier to handle and prepare it for further factoring steps. This method reduces the complexity of the expression and often reveals simpler forms of equations that can be easily solved.

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. When dealing with a polynomial equation, particularly those that are quadratic or higher degree, factoring is a powerful method to find the solutions.
After factoring, the equation \( 2x(x - 3)^2 = 0 \) consists of terms that are multiplied. According to the zero product property, if the product of factors equals zero, then at least one of the factors must be zero.
To solve

• \( 2x = 0 \) gives us \( x = 0 \)
• \((x - 3)^2 = 0\) leads to \( x - 3 = 0 \), so \( x = 3 \)

Thus, the solutions are \( x = 0 \) and \( x = 3 \).
The goal is to break down the polynomial into easily manageable factors, making it simpler to find solutions to the equation. Using factoring techniques efficiently can help resolve even more complex equations swiftly.