The given equation is:

\[
y^2 - 2x = 1
\]

To solve for \( y^2 \), we can rearrange the equation:

\[
y^2 = 2x + 1
\]

### Steps:
1. Rearrange the equation to isolate \( y^2 \).
2. If needed, you can solve for \( y \) by taking the square root:

\[
y = \pm \sqrt{2x + 1}
\]

This gives two possible values for \( y \) (positive and negative roots), depending on \( x \).

Would you like a more detailed explanation, or perhaps solve for specific values of \( x \)?

Here are five related questions:
1. What happens if you substitute a specific value for \( x \)?
2. Can this equation represent a conic section (e.g., parabola)?
3. How does the graph of this equation behave for negative values of \( x \)?
4. How would this equation change if it were \( y^2 - 2x = 0 \)?
5. Can we apply calculus to this equation to find the derivative of \( y \) with respect to \( x \)?

**Tip:** When solving equations with square roots, always remember to consider both the positive and negative roots.

Learn how to solve the equation y^2 - 2x = 1 using algebraic methods and square root properties. This solution demonstrates how to isolate y^2 and solve for y with step-by-step guidance. Suitable for students in Grades 9-10.

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