Problem 24 - Elementary Algebra

Question

Solve. $$ (x-6)(x-6)=0 $$

Step-by-Step Solution

Answer

x = 6

Step 1: Identify the equation

The given equation is \( (x-6)(x-6)=0 \). Notice that both factors are identical.

Step 2: Apply the Zero-Product Property

The Zero-Product Property states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). Here, we have \( (x-6)(x-6)=0 \), so set \( (x-6)=0 \).

Step 3: Solve for x

To solve \( x - 6 = 0 \), simply add 6 to both sides: \[ x - 6 + 6 = 0 + 6 \] \[ x = 6 \]

Key Concepts

Zero-Product Property

To solve the given equation, we first use the Zero-Product Property. This property is crucial when dealing with quadratic equations in the form of factors.
The Zero-Product Property states that if the product of two expressions is zero, then at least one of the factors must be zero.
For instance, if we have an equation like \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\).
In this exercise, the equation is \( (x-6)(x-6)=0 \). Applying the Zero-Product Property, we need to set each factor to zero. But here, both factors are the same, so we only need to solve \( (x-6)=0 \) once. This is an efficient way of solving the equation.

Factoring

Next, let's understand why factoring is important in solving quadratic equations. Factoring is the process of breaking down a complicated expression into simpler components (factors) that can be multiplied together to get the original expression.
Factoring is useful in many areas of algebra because it makes solving equations easier.
For example, consider the expression \(x^2 - 12x + 36\). It can be factored into \((x-6)(x-6)\).
This step is essential because once we have the equation in a factored form, we can then use the Zero-Product Property to solve for the variable.
For our particular problem, we started with \( (x-6)(x-6)=0 \), which is already factored, so applying the Zero-Product Property directly becomes straightforward.

Basic Algebra

Basic algebra skills are the foundation for understanding and solving quadratic equations.
In this problem, we utilize these skills when we solve \(x - 6 = 0\).
To isolate \(x\), we need to perform simple algebraic manipulations.
Here’s how:

• Add 6 to both sides of the equation: \(x - 6 + 6 = 0 + 6\)
• Simplify both sides to get: \(x = 6\)

Thus, the solution is \(x = 6\). Using basic algebra like addition, subtraction, and simplification makes solving such equations manageable.
Knowing these skills will help you tackle more complex algebraic equations effectively.