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Algebraic identities · 3 marks

Find the values of the following using the identity (a − b)2 = a2 − 2ab + b2.

  1. (i) (79)2
  2. (ii) (193)2
  3. (iii) (299)2
Answer: (i) 792 = 6241 (ii) 1932 = 37249 (iii) 2992 = 89401

Step-by-step solution

Idea: Each number is just below a round number: 79 = 80 − 1, 193 = 200 − 7, 299 = 300 − 1. Write it as (round number − small number) and use (a − b)2 = a2 − 2ab + b2. Only the middle term is subtracted; b2 is always added.

(i) (79)2

  1. 79 = 80 − 1, so a = 80 and b = 1.
  2. (79)2 = (80 − 1)2 = 802 − 2(80)(1) + 12½ mark
  3. = 6400 − 160 + 1 = 6241½ mark
6241

(ii) (193)2

  1. 193 = 200 − 7, so a = 200 and b = 7.
  2. (193)2 = (200 − 7)2 = 2002 − 2(200)(7) + 72½ mark
  3. = 40000 − 2800 + 49 = 37200 + 49 = 37249½ mark
37249

(iii) (299)2

  1. 299 = 300 − 1, so a = 300 and b = 1.
  2. (299)2 = (300 − 1)2 = 3002 − 2(300)(1) + 12½ mark
  3. = 90000 − 600 + 1 = 89401½ mark
89401
(i) (79)² = 6241 (ii) (193)² = 37249 (iii) (299)² = 89401

Check: Last digits: 9 × 9 = 81 ends in 1, so 792 and 2992 must end in 1 ✓; 3 × 3 = 9, so 1932 must end in 9 ✓. Size: 193 is a little under 200, and 37249 is a little under 40000 ✓.

Answer to write in the exam

(i)

(79)2 = (80 − 1)2 = 802 − 2(80)(1) + 12 [(a − b)2 = a2 − 2ab + b2]

= 6400 − 160 + 1

∴ (79)2 = 6241

(ii)

(193)2 = (200 − 7)2 = 2002 − 2(200)(7) + 72 [(a − b)2 = a2 − 2ab + b2]

= 40000 − 2800 + 49

∴ (193)2 = 37249

(iii)

(299)2 = (300 − 1)2 = 3002 − 2(300)(1) + 12 [(a − b)2 = a2 − 2ab + b2]

= 90000 − 600 + 1

∴ (299)2 = 89401

Common mistakes that cost marks

  • Subtracting b2 as well: writing 6400 − 160 − 1 = 6239. In (a − b)2 the last term is +b2, because (−b) × (−b) is positive.
  • Writing (80 − 1)2 = 802 − 12 = 6399. The middle term −2ab = −160 is missing.
  • Choosing a split that makes the work harder, such as 193 = 190 + 3. It works, but 200 − 7 is quicker because 2002 is easy.

How this can come in the exam

MCQ (1 mark)

Using (a − b)2 = a2 − 2ab + b2, the value of (98)2 is

  1. 9404
  2. 9804
  3. 9604
  4. 9614
Show answer

(C) 9604
(100 − 2)2 = 10000 − 400 + 4 = 9604.

Assertion–Reason (1 mark)

Assertion (A): (49)2 = 2401.
Reason (R): (50 − 1)2 = 502 − 2(50)(1) + 12.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
R is the identity (a − b)2 with a = 50, b = 1, and it gives 2500 − 100 + 1 = 2401, which is A.

Try one yourself

Using (a − b)2 = a2 − 2ab + b2, find (999)2 and (88)2.

Show answer

(1000 − 1)2 = 1000000 − 2000 + 1 = 998001; (90 − 2)2 = 8100 − 360 + 4 = 7744.

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