Exploring the Solutions to Quadratic Equations: n2 - 1 1

Understanding the Problem

The problem presented is: for what value of n is ( n^2 - 1 - 1 0 )? To solve this, we first simplify the equation to ( n^2 - 1 - 1 0 ) which can be further simplified to ( n^2 - 2 0 ). By adding 2 to both sides, we obtain ( n^2 2 ).

Breaking Down the Solution

Let's start by analyzing the simplified equation ( n^2 - 2 0 ) to find the values of n that satisfy it.

Case 1: ( n^2 - 2 1 )

For this case, we are solving ( n^2 - 2 1 ), which simplifies to ( n^2 3 ). However, as no value of n will make ( n^2 3 ) hold true in the context of the original equation ( n^2 - 1 - 1 0 ), this case is not relevant.

Case 2: ( n^2 - 1 1 )

In this scenario, we solve ( n^2 - 1 1 ). Simplifying, we get ( n^2 2 ). The values of n that satisfy this equation are ( n sqrt{2} ) and ( n -sqrt{2} ).

Case 3: ( n^2 - 1 -1 )

Here, we address the equation ( n^2 - 1 -1 ), which simplifies to ( n^2 0 ). The only value of n that satisfies this equation is ( n 0 ).

Combining the Results

By combining the results from the above cases, we find that the solutions are ( n sqrt{2} ), ( n -sqrt{2} ), and ( n 0 ).

Conclusion

The correct answer is therefore D: I and II and III. This means that n can be (sqrt{2}), (-sqrt{2}), or 0, depending on the original given equation and how the terms are handled.

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This article not only explores the solution to a specific mathematical problem but also provides an explanation and reasoning that can help improve understanding and retention of the concept.