4a = 2b^2 \implies 2a = b^2 \implies b^2 = 2a

The sequence of equations you've provided is a valid manipulation of the original equation. Here's the step-by-step breakdown:

1. **Original equation:**
   \[
   4a = 2b^2
   \]

2. **Divide both sides by 2:**
   \[
   \frac{4a}{2} = \frac{2b^2}{2}
   \]
   This simplifies to:
   \[
   2a = b^2
   \]

3. **Rearranging the equation:**
   \[
   b^2 = 2a
   \]

So, the final expression \( b^2 = 2a \) correctly follows from the original equation.

Would you like further details or have any questions about these steps? Here are some related questions you might find useful:

1. What are some potential applications of the equation \( b^2 = 2a \)?
2. How would you solve for \( b \) in terms of \( a \) from the equation \( b^2 = 2a \)?
3. What does this equation suggest about the relationship between \( a \) and \( b \)?
4. How could you use this equation to analyze a quadratic function?
5. What is the geometric interpretation of the equation \( b^2 = 2a \) in a coordinate plane?

**Tip:** When solving equations, it's often helpful to check each step by substituting values back into the original equation to ensure consistency.

Explore how to solve the equation 4a = 2b^2 step-by-step with detailed explanations. This problem involves equations and quadratic equations, discussing key concepts and formulas. Suitable for high school students and beyond.

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