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Solving Equations Using the Zero Product Property
The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is extremely helpful in solving equations, particularly those that are in factored form. Let's take a look at how this works in practice with the equation (x-9)(x-8) = 0.
Understanding the Equation
We have the product of two factors, (x-9) and (x-8), that equal zero. The Zero Product Property tells us that for this to be true, either (x-9) must equal zero or (x-8) must equal zero.
Solving for x
To find the solutions, we set each factor equal to zero and solve:
- (x-9) = 0
- Add 9 to both sides: x = 9
- (x-8) = 0
- Add 8 to both sides: x = 8
Therefore, the solutions to the equation (x-9)(x-8) = 0 are x = 9 and x = 8.
Verification
We can verify our solutions by plugging them back into the original equation:
- For x = 9: (9-9)(9-8) = 0 * 1 = 0 (True)
- For x = 8: (8-9)(8-8) = -1 * 0 = 0 (True)
Both solutions satisfy the original equation, confirming our findings.
Conclusion
The Zero Product Property provides a straightforward method for solving equations in factored form. By setting each factor equal to zero and solving, we can determine all possible solutions. This property is a fundamental tool in algebra and has wide applications in various mathematical contexts.