(n%2B1)%3D0.jpeg "(5n-1)(n+1)=0")
Solving the Equation (5n-1)(n+1) = 0
This equation is a quadratic equation in the form of a product of two linear factors equaling zero. This allows us to use the Zero Product Property to easily solve for the possible values of n.
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.
Applying the Property
In our equation, (5n-1)(n+1) = 0, we have two factors: (5n-1) and (n+1).
For the product to be zero, at least one of these factors must be zero. Therefore, we set each factor equal to zero and solve for n:
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Factor 1: 5n - 1 = 0
- Add 1 to both sides: 5n = 1
- Divide both sides by 5: n = 1/5
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Factor 2: n + 1 = 0
- Subtract 1 from both sides: n = -1
Solutions
Therefore, the solutions to the equation (5n-1)(n+1) = 0 are n = 1/5 and n = -1.
Verification
We can verify our solutions by substituting them back into the original equation:
- For n = 1/5: (5(1/5) - 1)(1/5 + 1) = (1 - 1)(6/5) = 0 * 6/5 = 0
- For n = -1: (5(-1) - 1)(-1 + 1) = (-6) * 0 = 0
Both solutions satisfy the original equation, confirming their validity.