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The given equation involves two expressions that we are asked to sum. Let's break down both sides and then combine them step by step.

The expression is:

\[
\frac{1}{2}a^2 - \frac{3}{4}b^2 + \frac{3}{4}a^2 + \frac{5}{6}b^2 - \frac{2}{5}
\]

### Steps to combine like terms:

1. **Group terms with \( a^2 \):**
   - \(\frac{1}{2}a^2 + \frac{3}{4}a^2\)

   We need a common denominator for these fractions:
   \[
   \frac{1}{2}a^2 = \frac{2}{4}a^2
   \]
   So:
   \[
   \frac{2}{4}a^2 + \frac{3}{4}a^2 = \frac{5}{4}a^2
   \]

2. **Group terms with \( b^2 \):**
   - \(-\frac{3}{4}b^2 + \frac{5}{6}b^2\)

   Again, we need a common denominator:
   \[
   \text{LCM of 4 and 6 is 12.}
   \]
   Rewriting both fractions with denominator 12:
   \[
   -\frac{3}{4}b^2 = -\frac{9}{12}b^2, \quad \frac{5}{6}b^2 = \frac{10}{12}b^2
   \]
   So:
   \[
   -\frac{9}{12}b^2 + \frac{10}{12}b^2 = \frac{1}{12}b^2
   \]

3. **The constant term remains:**
   - \(-\frac{2}{5}\)

### Final expression:
After summing the terms, the final result is:

\[
\frac{5}{4}a^2 + \frac{1}{12}b^2 - \frac{2}{5}
\]

Would you like a further breakdown or more details?

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Here are 5 related questions for further exploration:
1. How would the result change if one of the constants (like \(\frac{2}{5}\)) was a variable?
2. Can this expression be simplified further if specific values for \(a\) and \(b\) are given?
3. What would the steps be if you had to solve for \(a\) or \(b\) given specific values?
4. How does the result change if different powers of \(a\) or \(b\) are involved, such as \(a^3\) or \(b^3\)?
5. Can you identify the significance of least common denominators in simplifying expressions?

**Tip:** When adding algebraic fractions, always make sure to find the least common denominator (LCD) to combine them efficiently!

Sum the expressions: \(\frac{1}{2}a^2 - \frac{3}{4}b^2\) and \(\frac{3}{4}a^2 + \frac{5}{6}b^2 - \frac{2}{5}\).

Learn how to sum algebraic expressions involving fractional coefficients and powers of variables, with a detailed step-by-step solution. This problem covers key algebra concepts such as combining like terms and working with least common denominators. Suitable for students in Grades 9-11.

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