Solving Quadratic Equations Using the Zero Product Property

In this article, we will explore how to solve a specific type of quadratic equation using the zero product property. Quadratic equations are fundamental in algebra, and they are expressed in the form (ax^2 bx c 0). The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is particularly useful in solving equations that can be factored.

Example: Solving the Equation 12x^2 - 9x - 6 6

Let's take the equation 12x^2 - 9x - 6 6 and solve it step by step.

Step 1: Simplify the Equation

We start by moving all terms to one side of the equation to set it equal to zero:

12x^2 - 9x - 6 - 6  0

This simplifies to:

12x^2 - 9x  0

Step 2: Factor the Equation

The next step is to factor the quadratic expression. We can factor out the greatest common factor (GCF) from the terms:

3x(4x - 3)  0

Step 3: Apply the Zero Product Property

Using the zero product property, if the product of two factors is zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:

3x 0 implies x 0 4x - 3 0 implies 4x 3, and x frac{3}{4}

Hence, the possible solutions for x are 0 and frac{3}{4}.

Summary of the Steps

The process of solving the quadratic equation 12x^2 - 9x - 6 6 can be summarized as follows:

Transform the equation to 12x^2 - 9x 0 by subtracting 6 on both sides.

Factor the equation to get 3x(4x - 3) 0.

Apply the zero product property to find the values of x: x 0 or x frac{3}{4}.

Additional Examples

Let's look at a few more examples to further illustrate the concept:

Example 1:

12x^2 - 9x - 6 - 6 6 - 6

This simplifies to:

12x^2 - 9x 0

From here, we factor and apply the zero product property to find the solutions for x.

Example 2:

12x^2 - 9x - 6 6

12x^2 - 9x - 6 - 6 6 - 6

12x^2 - 9x 0

3x(4x - 3) 0

Using the factor and zero product properties, we get x 0 or x frac{3}{4}.

Example 3: