Problem 11 - Elementary Algebra

Question

Solve. $$ (x-2) 2=0 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Understand the equation

The equation provided is \((x - 2) \cdot 2 = 0\). This equation implies that the expression \((x - 2)\) is being multiplied by 2, and the product is equal to zero.

Step 2: Use the Zero Product Property

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Here, we can divide both sides of the equation by 2 to isolate \((x - 2)\), resulting in \((x - 2) = 0\).

Step 3: Solve for x

Now solve the equation \((x - 2) = 0\) for \(x\). Add 2 to both sides of the equation to obtain \(x = 2\).

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that helps in solving equations involving products. It asserts that if the product of two numbers (or expressions) is zero, then at least one of those numbers must be zero. Imagine multiplying two things and getting zero as the result. It can only happen if one or both things you're multiplying are themselves zero.

For example, if you have an equation like

• \((a) \cdot (b) = 0\)

you can conclude that either

• \(a = 0\)
• or \(b = 0\)

Applying this property often simplifies solving equations, especially when one or more terms in the equation are set to zero. Understanding and applying the Zero Product Property can make solving many algebraic problems much easier.

Isolating the Variable

Isolating the variable is a crucial step in solving equations. It simply means rearranging the equation to get the variable alone on one side. This helps you find the value of the variable that satisfies the equation.

In the equation

• \((x - 2) \cdot 2 = 0\)

you can first divide both sides by 2 to simplify and isolate \((x - 2)\), resulting in

• \((x - 2) = 0\)

Once the variable is isolated, it becomes much easier to solve for its value. Think of it as unwrapping a present to find the gift inside. By isolating the variable, you are peeling away other numbers or operations wrapped around it. This process of isolating ensures clarity and simplicity in solving equations.

Solution Steps for Equations

To solve equations effectively, a series of logical steps need to be followed, making the process consistent and straightforward. Let’s go through these steps using an example from the provided exercise.

• **Understanding the Problem:** Start by recognizing what is being asked. The equation \((x - 2) \cdot 2 = 0\) involves finding the value of \(x\) that makes the equation true.
• **Using the Zero Product Property:** Following this principle, divide both sides of the equation by 2, giving you \((x - 2) = 0\). This simplification helps focus only on the important parts of the equation.
• **Solving for the Variable:** With \((x - 2) = 0\), the next step is to add 2 to both sides to solve for \(x\). This simple addition leads us to \(x = 2\).

These steps highlight important techniques like understanding, applying properties, and isolating variables, all crucial for solving any equation efficiently. Mastering these steps can boost confidence and accuracy in algebra.