The equation provided is:

\[
v^2 - 1 = 1
\]

To solve for \( v \), follow these steps:

1. Add 1 to both sides of the equation to isolate the \( v^2 \) term:
   \[
   v^2 = 2
   \]

2. Take the square root of both sides to solve for \( v \):
   \[
   v = \pm\sqrt{2}
   \]

So, the solution is \( v = \sqrt{2} \) or \( v = -\sqrt{2} \).

Would you like a detailed explanation or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of taking the square root when solving quadratic equations?
2. How do you solve equations that involve higher powers, such as \( v^3 - 1 = 1 \)?
3. Can you solve the equation \( v^2 + 1 = 0 \)? What does the solution represent?
4. How does the discriminant help in determining the nature of solutions for quadratic equations?
5. What is the process for solving quadratic equations by completing the square?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Explore how to solve the quadratic equation v^2 - 1 = 1 step-by-step. This solution covers important algebraic concepts such as isolating variables and applying the square root property, suitable for students in Grades 8-10.

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