Simplify the expression (5p−6q)² + (5p+6q)²

The correct answer is 50p² + 72q². This algebraic simplification uses the identities for the square of a binomial: (a - b)² = a² - 2ab + b² and (a + b)² = a² + 2ab + b². When added, the middle terms cancel beautifully.

Step-by-Step Solution

• Expand the First Square (5p - 6q)²:
  
  Using (a - b)² = a² - 2ab + b², with a = 5p and b = 6q:
  
  (5p)² - 2×(5p)×(6q) + (6q)²
  
  = 25p² - 60pq + 36q²
• Expand the Second Square (5p + 6q)²:
  
  Using (a + b)² = a² + 2ab + b²:
  
  (5p)² + 2×(5p)×(6q) + (6q)²
  
  = 25p² + 60pq + 36q²
• Add the Two Expanded Expressions:
  
  (25p² - 60pq + 36q²) + (25p² + 60pq + 36q²)
  
  Combine like terms:
  
  25p² + 25p² = 50p²
  
  -60pq + 60pq = 0 (the middle terms cancel out)
  
  36q² + 36q² = 72q²
  
  Result: 50p² + 72q²
• Verification:
  
  Let p = 1, q = 1. Original expression: (5-6)² + (5+6)² = (-1)² + (11)² = 1 + 121 = 122.
  
  Our result: 50(1)² + 72(1)² = 50 + 72 = 122. Matches.

Thus, the simplified expression is 50p² + 72q².