Problem 21 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ (3 a+1)(3 a-1)=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the given equation are \(a = \frac{-1}{3}\) and \(a = \frac{1}{3}\).

Step 1: Set the first factor to zero

Set \((3a+1) = 0\). This will give us our first solution for \(a\).

Step 2: Solve for a using the first factor

Solving the equation \((3a+1) = 0\), we obtain: \[ 3a = -1 \implies a = \frac{-1}{3} \]

Step 3: Set the second factor to zero

Set \((3a-1) = 0\). This will give us our second solution for \(a\).

Step 4: Solve for a using the second factor

Solving the equation \((3a-1) = 0\), we obtain: \[ 3a = 1 \implies a = \frac{1}{3} \]

Step 5: Combine the solutions

Both solutions for \(a\) are valid since the equation is a quadratic equation. Therefore, the solutions of the given equation are: \[ a = \frac{-1}{3} \; \text{or} \; a = \frac{1}{3} \]

Key Concepts

Zero Product Property

The Zero Product Property is a simple yet powerful tool used to solve quadratic equations. It states that if the product of two expressions equals zero, then at least one of the expressions must be zero. Formally, if \(ab = 0\), then either \(a = 0\), \(b = 0\), or both. This property is essential for solving polynomial equations that have been factored into a product of simpler expressions.

In our exercise with the equation \((3a+1)(3a-1) = 0\), you apply the Zero Product Property by setting each factor equal to zero. This leads to two separate equations: \(3a+1=0\) and \(3a-1=0\). By solving these, you find the values of \(a\) that satisfy the original equation.

Factoring

Factoring is the process of breaking down an expression into the product of simpler terms, or factors. It is a crucial step in solving quadratic equations when they can be expressed in the form \( (ax+b)(cx+d)=0 \). It's like defragmenting a number into its prime components, but for equations.

In our example, the expression \((3a+1)(3a-1)\) is already factored. This simple multiplication tricks is known as the difference of squares, which results in the original quadratic form. Factoring sets the stage for using the Zero Product Property by providing a format where this property can be applied.

This method is not just limited to equations like the ones in the exercise, but can also be used for more complex polynomials. Recognizing patterns, such as differences of squares and grouping of like terms, allows for an efficient approach to finding solutions.

Solution of Equations

Solving equations involves finding the values of the variable that make the equation true. When dealing with quadratic equations, factoring combined with the Zero Product Property is an effective strategy.

For the equation \((3a+1)(3a-1)=0\), the solution of the equation comes from solving each factor after setting it to zero. Through basic algebraic manipulation, each factor's equation is solved:

• \(3a+1=0\) becomes \(3a = -1\), so \(a = \frac{-1}{3}\).
• \(3a-1=0\) becomes \(3a = 1\), so \(a = \frac{1}{3}\).

Thus, the solutions to the equation are \(a = \frac{-1}{3}\) and \(a = \frac{1}{3}\).

These represent the values of \(a\) where the original equation holds true. Solving equations in this manner lays down the foundation for tackling more complex algebraic problems.