Problem 11 - Algebra A Combined Function

Question

Solve each equation. $$ 2 x(x+12)=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = -12 \).

Step 1: Identify the Solutions for Product Zero

We start by understanding that for the equation \( 2x(x+12) = 0 \) to be true, at least one of the factors must be zero. This is because if any factor of a product is zero, the entire product is zero.

Step 2: Set Each Factor to Zero

The equation is factored as \( 2x(x + 12) = 0 \). We have two factors: \( 2x \) and \( (x + 12) \). Set each factor equal to zero.1. \( 2x = 0 \)2. \( x + 12 = 0 \)

Step 3: Solve Each Factor Equation

Solve each of the simple equations obtained from setting each factor to zero.1. Solve \( 2x = 0 \): Divide both sides by 2 to get \( x = 0 \).2. Solve \( x + 12 = 0 \): Subtract 12 from both sides to obtain \( x = -12 \).

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental concept in algebra, especially when dealing with quadratic equations and factoring. It states that if the product of two or more factors is zero, at least one of the factors must be zero. This property can be incredibly useful in solving quadratic equations. For example, in the equation \[ 2x(x+12) = 0, \]we have a product of two factors: \(2x\) and \(x + 12\). According to the Zero Product Property, at least one of these factors must equal zero to satisfy the equation.

• If \(2x = 0\), then one solution for \(x\) is found by dividing both sides by 2, resulting in \(x = 0\).
• If \(x + 12 = 0\), we find another solution by subtracting 12 from both sides, which gives \(x = -12\).

This property simplifies solving complex equations by allowing them to be broken down into simpler, more manageable parts.

Factoring

Factoring is the process of expressing a polynomial as a product of its factors. In the given problem, the expression \(2x(x+12)\) is already factored for us, but it is important to understand what that means. The process of factoring usually involves looking for common factors or using special techniques for polynomials, such as grouping or using the quadratic formula when applicable. Here, we have a polynomial \(2x(x + 12)\), which is written as the product of \(2x\) and \(x + 12\).

• Common Factor: Often, expressions can be simplified by factoring out the greatest common factor (GCF).
• Difference of Squares: Polynomials of the form \(a^2 - b^2\) can be factored as \((a+b)(a-b)\).
• Quadratic Binomials: The basic quadratic expression \(x^2 + bx + c\) can be factored using methods like completing the square or the quadratic formula.

Factoring transforms complex expressions into multiple simpler parts, making them easier to solve when set equal to zero, as we do in equations.

Solving Linear Equations

Solving linear equations is one of the foundational skills in algebra. After factoring an equation and using the Zero Product Property, you often end up with linear equations to solve, like we did in our problem:\[2x = 0 \quad \text{and} \quad x + 12 = 0.\]Solving these linear equations involves performing basic operations to isolate \(x\), the variable. Let's briefly recap these steps:

• Isolate the Variable: You want to get \(x\) by itself on one side of the equation.
• Use Inverse Operations: If \(2x = 0\), divide both sides by 2 to find that \(x = 0\).
• Balance the Equation: In \(x + 12 = 0\), subtract 12 from both sides to achieve \(x = -12\).

Breaking down each step ensures precision and reduces errors. Practicing these strategies will boost your confidence in handling more complex equations in the future.