6.4 Mind the Mistake, Mend the Mistake

Spotting a wrong step is one of the best ways to understand a right one. Each “simplification” below has been attempted. Decide whether it is correct; if not, find the slip and write the correct expression.

Detective Work

  1. -3p(-5p+2q) = -3p + 5p - 2q = p - 2q
  2. 2(x-1) + 3(x+4) = 2x - 1 + 3x + 4 = 5x + 3
  3. y + 2(y+2) = (y+2)^2 = y^2 + 4y + 4
  4. (5m+6n)^2 = 25m^2 + 36n^2
  5. (-q+2)^2 = q^2 - 4q + 4
  6. 3a(2b\times3c) = 6ab \times 9ac = 54a^2bc
  7. \dfrac{1}{2}(10s-6) + 3 = 5s - 3 + 3 = 5s
  8. 5w^2 + 6w = 11w^2
  9. 2a^3 + 3a^3 + 6a^2b + 6ab^2 = 5a^3 + 12a^2b^2
  10. (x+2)(x+5) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10
  11. (a+2)(b+4) = ab + 8
  12. ab^2 + a^2b + a^2b^2 = ab(a + b + ab)
  1. Wrong. Distributing -3p gives (-3p)(-5p) + (-3p)(2q) = 15p^2 - 6pq. The student forgot to multiply the p’s. Correct: \mathbf{15p^2 - 6pq}.
  2. Wrong. 2(x-1) = 2x - 2, not 2x - 1. Correct: 2x - 2 + 3x + 12 = \mathbf{5x + 10}.
  3. Wrong. y + 2(y+2) = y + 2y + 4 = 3y + 4. It was not a square at all. Correct: \mathbf{3y + 4}.
  4. Wrong. (5m+6n)^2 needs the middle term: 25m^2 + 2(5m)(6n) + 36n^2. Correct: \mathbf{25m^2 + 60mn + 36n^2}.
  5. Correct. (-q+2)^2 = (2-q)^2 = q^2 - 4q + 4. ✓
  6. Wrong. Inside, 2b \times 3c = 6bc, so 3a(6bc) = 18abc. The student wrongly split the product. Correct: \mathbf{18abc}.
  7. Correct. \tfrac12(10s-6) = 5s - 3, then +3 gives 5s. ✓
  8. Wrong. 5w^2 and 6w are not like terms (different powers of w). They cannot be combined. Correct: \mathbf{5w^2 + 6w}.
  9. Wrong. Two slips: 6a^2b and 6ab^2 are unlike, so they stay separate, and the exponent on the last term must not change. Correct: \mathbf{5a^3 + 6a^2b + 6ab^2}.
  10. Correct. x^2 + 7x + 10. ✓
  11. Wrong. Full expansion: ab + 4a + 2b + 8. Correct: \mathbf{ab + 4a + 2b + 8}.
  12. Correct. Factoring out ab: ab(a + b + ab). ✓ (Expand to check.)