Expanding Binomial Products — Solutions

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1. Identify and apply special product rules

   1. (x + 5)²: Perfect square (sum): x² + 10x + 25
   2. (y − 4)²: Perfect square (difference): y² − 8y + 16
   3. (a + 8)(a − 8): Difference of two squares: a² − 64
   4. (2x + 3)²: Perfect square (sum): 4x² + 12x + 9
   5. (3y − 7)²: Perfect square (difference): 9y² − 42y + 49
   6. (5m + 2)(5m − 2): Difference of two squares: 25m² − 4
   7. (4p − 9)²: Perfect square (difference): 16p² − 72p + 81
   8. (6k + 1)²: Perfect square (sum): 36k² + 12k + 1
   9. (x + 11)(x − 11): Difference of two squares: x² − 121
   10. (7a − 3)(7a + 3): Difference of two squares: 49a² − 9

2. Expand and simplify using FOIL

   1. (x + 2)(x + 9): x² + 11x + 18
   2. (a − 5)(a + 3): a² − 2a − 15
   3. (2m + 7)(m − 4): 2m² − m − 28
   4. (3n − 2)(n + 6): 3n² + 16n − 12
   5. (4x + 5)(2x − 1): 8x² + 6x − 5
   6. (3a − 4)(3a − 4): 9a² − 24a + 16
   7. (5y + 2)(y − 3): 5y² − 13y − 6
   8. (2p − 9)(3p + 1): 6p² − 25p − 9

3. Verify by substitution

   1. (x + 3)(x − 3) verification: x² − 9. Check x=5: original = 8×2 = 16; answer = 25−9 = 16 ✓
   2. (2a + 1)² verification: 4a² + 4a + 1. Check a=3: original = 7² = 49; answer = 36+12+1 = 49 ✓
   3. (4m − 2)(4m + 2) verification: 16m² − 4. Check m=2: original = 6×10 = 60; answer = 64−4 = 60 ✓
   4. (x + y)² ≠ x² + y²: (x+y)² = x² + 2xy + y². Sub x=3, y=4: LHS = 49; x²+y² = 9+16 = 25 ≠ 49. The missing term is 2xy = 2(3)(4) = 24.

4. Reasoning with special products

   1. 49 × 51: (50−1)(50+1) = 50² − 1² = 2500 − 1 = 2499
   2. 102²: (100+2)² = 10000 + 400 + 4 = 10404
   3. (x−3)² vs (3−x)²: (x−3)² = x²−6x+9 and (3−x)² = 9−6x+x² = x²−6x+9. They are equal because (x−3) = −(3−x) and squaring removes the negative.
   4. (a+b)² = a²+b² when?: Only when ab = 0, i.e. when a = 0 or b = 0 (or both). The middle term 2ab must be zero.

5. Applications of binomial products

   1. Square garden with path:
      1. Total area: Outer side = (x+4)+2 = x+6. Total area = (x+6)² = x² + 12x + 36 m²
      2. Path area expression: (x+6)² − (x+4)²
      3. Simplified path area: (x²+12x+36) − (x²+8x+16) = 4x + 20 m²
   2. Difference of squares simplified: (n+7)² = n²+14n+49 and (n−7)² = n²−14n+49. Difference = 28n. The result is always a multiple of 28n — a linear expression.
   3. Swimming pool area: (3x+2)(3x−2) = 9x²−4 m². When x=5: 9(25)−4 = 225−4 = 221 m²

6. Identities for mental arithmetic

   1. 49 × 51 = (50 − 1)(50 + 1) = 50² − 1 = 2499
   2. 102² = (100 + 2)² = 10 000 + 400 + 4 = 10 404
   3. 97² = (100 − 3)² = 10 000 − 600 + 9 = 9 409

7. Area expressions with binomial products

   1. Outer side = (x + 4) + 2 = x + 6 m
   2. (x + 6)² = x² + 12x + 36 m²
   3. Path area = (x + 6)² − (x + 4)²
   4. (x² + 12x + 36) − (x² + 8x + 16) = 4x + 20 m²

8. Reasoning with special products

   1. (n + 7)² = n² + 14n + 49
   2. (n − 7)² = n² − 14n + 49
   3. Difference = 28n. Always a multiple of 28n — a linear expression in n.
   4. (a + b)² = a² + b² only when 2ab = 0, i.e. when a = 0 or b = 0.

9. Swimming pool dimensions

   1. (3x + 2)(3x − 2) = 9x² − 4 m²
   2. x=5: 9(25) − 4 = 221 m²
   3. x=5: sides = 17 m and 13 m. P = 2(17 + 13) = 60 m
   4. 9x² − 4 ≥ 100 → x² ≥ 104/9 → x ≥ 3.4. Minimum integer x = 4 (check: 9(16)−4=140≥100 ✓)

10. Expanding beyond two brackets

    1. (x + 1)(x + 2) = x² + 3x + 2
    2. (x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6 = x³ + 6x² + 11x + 6
    3. x=1: original = 2×3×4 = 24; expanded = 1+6+11+6 = 24 ✓
    4. Degree = 3 (cubic)