Learnify Academy is a tuition centre in Bahrain. Classes are for students in Bahrain only.Tuition classes in Bahrain only

Algebraic identities · 3 marks

Using the same identity, find the values of the following:

  1. (i) (64)2
  2. (ii) (105)2
  3. (iii) (205)2
Answer: (i) 642 = 4096 (ii) 1052 = 11025 (iii) 2052 = 42025

Step-by-step solution

Idea: The identity is (a + b)2 = a2 + 2ab + b2. Split each number into a round number plus a small number (64 = 60 + 4, 105 = 100 + 5, 205 = 200 + 5). Round numbers and small numbers are easy to square in your head, so the identity turns a long multiplication into three easy ones.

(i) (64)2

  1. Write 64 = 60 + 4, so a = 60 and b = 4.
  2. (64)2 = (60 + 4)2 = 602 + 2(60)(4) + 42½ mark
  3. = 3600 + 480 + 16 = 4096½ mark
4096

(ii) (105)2

  1. Write 105 = 100 + 5, so a = 100 and b = 5.
  2. (105)2 = (100 + 5)2 = 1002 + 2(100)(5) + 52½ mark
  3. = 10000 + 1000 + 25 = 11025½ mark
11025

(iii) (205)2

  1. Write 205 = 200 + 5, so a = 200 and b = 5.
  2. (205)2 = (200 + 5)2 = 2002 + 2(200)(5) + 52½ mark
  3. = 40000 + 2000 + 25 = 42025½ mark
42025
(i) (64)² = 4096 (ii) (105)² = 11025 (iii) (205)² = 42025

Check: A quick test for (ii) and (iii): any number ending in 5 squares to a number ending in 25 ✓. For (i), 64 = 8 × 8, so 642 = 84 = 4096 ✓.

Answer to write in the exam

(i)

(64)2 = (60 + 4)2 = 602 + 2(60)(4) + 42 [(a + b)2 = a2 + 2ab + b2]

= 3600 + 480 + 16

∴ (64)2 = 4096

(ii)

(105)2 = (100 + 5)2 = 1002 + 2(100)(5) + 52 [(a + b)2 = a2 + 2ab + b2]

= 10000 + 1000 + 25

∴ (105)2 = 11025

(iii)

(205)2 = (200 + 5)2 = 2002 + 2(200)(5) + 52 [(a + b)2 = a2 + 2ab + b2]

= 40000 + 2000 + 25

∴ (205)2 = 42025

Common mistakes that cost marks

  • Writing (60 + 4)2 = 602 + 42 = 3616. The middle term 2ab = 480 has been left out.
  • Forgetting the 2 in 2ab: writing 100 × 5 = 500 instead of 2 × 100 × 5 = 1000 in (ii).
  • Slips with zeros: 2002 is 40000 (four zeros), not 4000.

How this can come in the exam

MCQ (1 mark)

Using (a + b)2 = a2 + 2ab + b2, the value of (103)2 is

  1. 10069
  2. 10900
  3. 10609
  4. 10309
Show answer

(C) 10609
(100 + 3)2 = 10000 + 600 + 9 = 10609.

Short answer (2 marks)

Find the value of (1002)2 using a suitable identity.

Show answer1002 = 1000 + 2. (1000 + 2)2 = 10002 + 2(1000)(2) + 22 = 1000000 + 4000 + 4 = 1004004. (½ mark for splitting, 1 mark for using the identity, ½ for the value.)

Try one yourself

Using (a + b)2 = a2 + 2ab + b2, find (52)2 and (301)2.

Show answer

(50 + 2)2 = 2500 + 200 + 4 = 2704; (300 + 1)2 = 90000 + 600 + 1 = 90601.

More questions like this

All Algebraic identities questions · All maths questions

← Using the identity (a + b)² = a² + 2ab + b², expand the following:The identity (a + b)² = a² + 2ab + b² can also be used to find factors →