Problem 71 - Algebra 2

Question

Solve each equation by factoring. \(4 x^{2}+8 x=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -2\).

Step 1: Identify the GCF

First, identify the greatest common factor (GCF) of all terms in the equation. For \(4x^2\) and \(8x\), the GCF is \(4x\).

Step 2: Factor Out the GCF

Factor out \(4x\) from the equation: \[ 4x(x + 2) = 0 \]

Step 3: Apply the Zero-Product Property

According to the zero-product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Apply this to our equation:\[ 4x = 0 \text{ or } x + 2 = 0 \]

Step 4: Solve Each Equation

Solve for \(x\) in each equation:1. From \(4x = 0\), divide both sides by 4:\[ x = 0 \]2. From \(x + 2 = 0\), subtract 2 from both sides:\[ x = -2 \]

Key Concepts

Greatest Common Factor

When we talk about the Greatest Common Factor (GCF) in algebra, we're looking for the largest factor that divides all terms in an equation without leaving a remainder. It’s like finding a common thread that ties the terms together. This step is crucial because simplifying is easier when you identify what they all share. For the equation \(4x^2 + 8x = 0\), we can see both terms, \(4x^2\) and \(8x\), include a common factor. Let's break it down:

• Find the GCF of the numerical coefficients 4 and 8, which is 4.
• Both terms include the variable \(x\). The smallest power of \(x\) shared is just \(x\), rather than \(x^2\) since \(8x\) lacks \(x^2\) entirely.

Thus, the GCF for this equation is \(4x\). Factoring this out simplifies the equation to focus on the relationship between the remaining terms. This initial step makes tackling the broader equation much more manageable.

Zero-Product Property

The Zero-Product Property is an essential concept in solving quadratic equations through factoring. It states that if a product of two or more terms equals zero, at least one of the terms must be zero. This is logical because only zero, when multiplied by any number, results in zero. In our factored equation \(4x(x + 2) = 0\):

• We apply the zero-product property, suggesting that either \(4x = 0\) or \(x + 2 = 0\).

The versatility of this property lies in breaking down complex equations into simpler, solvable components. It transforms the challenge into two separate, straightforward equations that can be addressed individually, leading us closer to finding the solution values for \(x\). By tackling each part singularly, we gain a clearer path to uncovering the roots of the equation. This property serves as a powerful tool in algebra, simplifying our journey towards understanding variable relationships.

Solving Equations

Solving equations, especially quadratic ones, often involves piecing together simpler puzzles to form a complete picture. Once we've factored the equation using the GCF and applied the zero-product property, the task of solving becomes a series of direct steps. Here's how we solve each part individually, as seen in the equation:

• For \(4x = 0\): Divide both sides by 4 to isolate \(x\), resulting in \(x = 0\).
• For \(x + 2 = 0\): Subtract 2 from both sides to solve for \(x\), giving us \(x = -2\).

Each branch of the equation provides a possible solution. In this case, our final solutions are \(x = 0\) and \(x = -2\). By breaking the problem into two smaller, easily solvable equations, we find the set of solutions. This method ensures we work methodically, validating the logical connections formed through the factoring and application of algebraic principles.