Problem 13 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ (x-6)(x-3)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the given quadratic equation are x = 6 and x = 3.

Step 1: Set Each Factor Equal to Zero separately

Since the product of the two factors is equal to zero, we know that either the first factor or the second factor (or both) must be equal to zero: $$ (x-6) = 0 $$ or $$ (x-3) = 0 $$

Step 2: Solve Each Equation Individually

Now, we will solve the two equations separately to find the solutions for x: For the first equation: $$ (x-6) = 0 \\ x = 6 $$ For the second equation: $$ (x-3) = 0 \\ x = 3 $$

Step 3: State the Solutions

The solutions for the given quadratic equation are x = 6 and x = 3. These are the values of x that make the equation true.

Key Concepts

Factoring

Factoring is an essential technique in solving quadratic equations. In the given example, \((x-6)(x-3) = 0\), the expression has already been factored. Factoring means writing the equation as a product of its factors, which are expressed as binomials in this case.

• Factoring transforms a polynomial into a multiplication problem of simpler expressions.
• This makes the solution easier because you only need to focus on each individual factor.

The goal is to express the quadratic equation in the form \((a+b)(c+d)\). Once in this form, the zero product property can be applied to each factor to find the solutions.
Understanding how to factor correctly is crucial to solving many algebraic equations.

Zero Product Property

The zero product property is a fundamental principle used when solving equations like \((x-6)(x-3) = 0\). This principle states that if the product of two factors is zero, then at least one of the factors must be zero.
For any numbers or expressions \(a\) and \(b\), if \(a \times b = 0\) then either \(a = 0\) or \(b = 0\) (or both).
Using this property, we can conclude:

• Since \((x-6)(x-3) = 0\), it follows that \(x-6 = 0\) or \(x-3 = 0\).
• This reduces the problem to solving each equation separately.

The zero product property simplifies the process of solving quadratic equations remarkably. It's particularly useful when the quadratic expression is neatly factored, as it directly leads to the solutions.

Algebraic Solutions

Finding algebraic solutions means solving for the variable, \(x\), to see what values satisfy the equation. In our example, after applying the zero product property, we solve each of the simpler equations separately:
1. For \(x-6 = 0\), solve by adding 6 to both sides to get \(x = 6\). 2. Similarly, for \(x-3 = 0\), add 3 to both sides yielding \(x = 3\).

• Both solutions \(x = 6\) and \(x = 3\), make the original equation true.
• These values collectively form the solution set for the quadratic equation.

Algebraic solutions neatly tie together the steps of factoring and using the zero product property, confirming that these methods lead us to all possible solutions of the equation. Understanding these solutions is key in comprehending how quadratic equations behave and how they model real-world scenarios.