(x-2)(x+7)=0 Zero Product Property

Jasonbradley

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Jun 17, 2024

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Solving Equations Using the Zero Product Property

The Zero Product Property is a powerful tool in algebra, allowing us to solve equations by factoring. It states:

If the product of two or more factors is zero, then at least one of the factors must be zero.

This property helps us find the solutions (also called roots) of equations where one side is factored and the other side is zero.

Example: (x - 2)(x + 7) = 0

Let's solve the equation (x - 2)(x + 7) = 0 using the Zero Product Property.

1. Identify the factors: The factors in this equation are (x - 2) and (x + 7).
2. Apply the Zero Product Property: According to the property, either (x - 2) = 0 or (x + 7) = 0.
3. Solve for x:
   • For (x - 2) = 0, add 2 to both sides to get x = 2.
   • For (x + 7) = 0, subtract 7 from both sides to get x = -7.

Therefore, the solutions to the equation (x - 2)(x + 7) = 0 are x = 2 and x = -7.

Understanding the Solution

We can check our solutions by substituting them back into the original equation:

• For x = 2: (2 - 2)(2 + 7) = 0 * 9 = 0, which is true.
• For x = -7: (-7 - 2)(-7 + 7) = -9 * 0 = 0, which is also true.

This confirms that our solutions are correct.

Summary

The Zero Product Property is a fundamental concept in algebra that allows us to solve factored equations efficiently. By setting each factor equal to zero and solving for the variable, we can determine the solutions or roots of the equation.