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1.

Expand:i. (2p + q + 5)2 ii. (m + 2n + 3r)2

Answer»

i. (2p + q + 5)2 = (2p)2 + (q)2 + (5)2 + 2(2p) (q) + 2(q) (5) + 2(2p) (5)

… [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac] 

= 4p2 + q2 + 25 + 4pq + 10q + 20p

ii. (m + 2n + 3r)2 = (m)2 + (2n)2 + (3r)2 + 2(m) (2n) + 2(2n) (3r) + 2(m) (3r)

… [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac]

= m2 + 4n2 + 9r2 + 4mn + 12nr + 6mr
Discussion
2.

Simplify:(x – 2y + 3)2 + (x + 2y – 3)2

Answer»

(x – 2y + 3)2 + (x + 2y – 3)2

= [(x)2 + (- 2y)2 + (3)2 + 2 (x) (- 2y) + 2 (- 2y) (3) + 2 (x) (3)] + [(x)2 + (2y)2 + (- 3)2 + 2 (x) (2y) + 2 (2y) (- 3) + 2 (x) (- 3)]

… [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac]

= x2 + 4y2 + 9 – 4xy – 12y + 6x + x2 + 4y2 + 9 + 4xy – 12y – 6x

= x + x2 + 4y2 + 4y2 + 9 + 9 – 4xy + 4xy – 12y – 12y + 6x – 6x

= 2x2 + 8y2 + 18 – 24y
Discussion
3.

Simplify:(3k – 4r – 2m)2 – (3k + 4r – 2m)2

Answer»

(3k – 4r – 2m)2 – (3k + 4r – 2m)2

= [(3k)2 + (- 4r)2 + (- 2m)2 + 2 (3k) (- 4r) + 2 (- 4r) (- 2m) + 2 (3k) (- 2m)] – [(3k)2 + (4r)2 + (- 2m)2 + 2 (3k) (4r) + 2 (4r) (- 2m) + 2 (3k) (- 2m)]

… [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac]

= (9k2 + 16r2 + 4m2 – 24kr + 16rm – 12km) – (9k2 + 16r2 + 4m2 + 24kr – 16rm – 12km)

= 9k2 + 16r2 + 4m2 – 24kr + 16rm – 12km – 9k2 – 16r2 – 4m2 – 24kr + 16rm + 12km

= 9k2 – 9k2 + 16r2 – 16r2 + 4m2 – 4m2 – 24kr – 24kr + 16rm + 16rm – 12km + 12km

= 32rm – 48kr
Discussion
4.

Simplify: i. (2a + b)3 – (2a – b)3ii. (3r – 2k)3 + (3r + 2k)3iii. (4a – 3)3 – (4a + 3)3iv. (5x – 7y)3 + (5x + 7y)3

Answer»

i. (2a + b)3 – (2a – b)3

= [(2a)3 + 3(2a)2(b) + 3 (2a)(b)2 + (b)3] – [(2a)3 – 3(2a)2(b) + 3 (2a)(b)2 – (b)3] … [(a + b)3 

= a3 + 3a2b + 3ab2 + b3, (a – b)3 = a3 – 3a2b + 3ab2 – b3] 

= (8a3 + 12a2b + 6ab2 + b3) – (8a3 – 12a2b + 6ab2 – b3) 

= 8a3 + 12a2b + 6ab2 + b3 – 8a3 + 12a2b – 6ab2 + b3 

= 8a3 – 8a3 + 12a2b + 12a2b + 6ab2 – 6ab2 + b3 + b3 

= 24a2b + 2b3

ii. (3r – 2k)3 + (3r + 2k)3 

= [(3r)3 – 3(3r)2(2k) + 3(3r)(2k)2 – (2k)3] + [(3r)3 + 3(3r)2(2k) + 3(3r)(2k)2 + (2k)3] … [(a – b)3 = a3 – 3a2b + 3ab2 – b3, (a + b)3= a3 + 3a2b + 3ab2 + b3] 

= (27r3 – 54r2k + 36rk2 – 8k3) + (27r3 + 54r2k + 36rk2 + 8k3) 

= 27r3 – 54r3k + 36rk2 – 8k3 + 27r3 + 54r2k + 36rk2 + 8k3

= 27r3 + 27r3 – 54r2k + 54r2k + 36rk2 + 36rk2 – 8k3 + 8k3 

= 54r3 + 72rk2

iii. (4a – 3)3 – (4a + 3)3 

= [(4a)3 – 3(4a)2 (3) + 3(4a)(3)2 – (3)3] – [(4a)3 + 3(4a)2(3) + 3(4a)(3)2 + (3)3] … [(a – b)3 = a3 – 3a2b + 3ab2 – b3, (a + b)3 = a3 + 3a2b + 3ab2 + b3] 

= (64a3 – 144a2 + 108a – 27) – (64a3 + 144a2 + 108a + 27) 

= 64a3 – 144a2 + 108a – 27 – 64a3 -144a2 – 108a – 27 

= 64a3 – 64a3 – 144a2 – 144a2 + 108a – 108a – 27 – 27 

= -288a2 – 54

v. (5x – 7y)3 + (5x + 7y)3 

= [(5x)3 – 3(5x)2(7y) + 3(5x)(7y)2 – (7y)3] + [(5x)3 + 3(5x)2 (7y) + 3(5x) (7y)2 +(7y)3] … [(a – b)3 = a3 – 3a2b + 3ab2 – b3, (a + b)3 = a3 + 3a2b + 3ab2 + b3] 

= (125x3 – 525x2y + 735xy2 – 343y3) + (125x3 + 525x2y + 735xy2 + 343y3) 

= 125x3 – 525x2y + 735xy2 – 343y3 + 125x3 + 525x2y + 735xy2 + 343y3 

= 125x3 + 125x3 – 525x2y + 525x2y + 735xy2 + 735xy2 – 343y3 + 343y3 

= 250x3 + 1470xy2
Discussion
5.

Simplify:(7a – 6b + 5c)2 + (7a + 6b – 5c)2

Answer»

(7a – 6b + 5c)2 + (7a + 6b – 5c)2 

= [(7a)2 + (- 6b)2 + (5c)2 + 2(7a) (-6b) + 2(-6b) (5c) + 2(7a) (5c)] + [(7a)2+ (6b)2 + (- 5c)2 + 2 (7a) (6b) + 2 (6b) (- 5c) + 2 (7a) (- 5c)]

… [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac] 

= 49a2 + 36b2 + 25c2 – 84ab – 60bc + 70ac + 49a2 + 36b2 + 25c2 + 84ab – 60bc – 70ac

= 49a2 + 49a2 + 36b2 + 36b2 + 25c2 + 25c2 – 84ab + 84ab – 60bc – 60bc + 70ac – 70ac

= 98a2 + 72b2 + 50c2 – 120bc
Discussion
6.

Simplify: i. (x – 2y + 3)2 + (x + 2y – 3)2ii. (3k – 4r – 2m)2 – (3k + 4r – 2m)2iii. (7a – 6b + 5c)2 + (7a + 6b – 5c)2

Answer»

i. (x – 2y + 3)2 + (x + 2y – 3)2

= [(x)2 + (- 2y)2 + (3)2 + 2 (x) (- 2y) + 2 (- 2y) (3) + 2 (x) (3)] + [(x)2 + (2y)2 + (- 3)2 + 2 (x) (2y) + 2 (2y) (- 3) + 2 (x) (- 3)] … [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac] 

= x2 + 4y2 + 9 – 4xy – 12y + 6x + x2 + 4y2 + 9 + 4xy – 12y – 6x 

= x + x2 + 4y2 + 4y2 + 9 + 9 – 4xy + 4xy – 12y – 12y + 6x – 6x 

= 2x2 + 8y2 + 18 – 24y

ii. (3k – 4r – 2m)2 – (3k + 4r – 2m)2 

= [(3k)2 + (- 4r)2 + (- 2m)2 + 2 (3k) (- 4r) + 2 (- 4r) (- 2m) + 2 (3k) (- 2m)] – [(3k)2 + (4r)2 + (- 2m)2 + 2 (3k) (4r) + 2 (4r) (- 2m) + 2 (3k) (- 2m)] … [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac] 

= (9k2 + 16r2 + 4m2 – 24kr + 16rm – 12km) – (9k2 + 16r2 + 4m2 + 24kr – 16rm – 12km)

= 9k2 + 16r2 + 4m2 – 24kr + 16rm – 12km – 9k2 – 16r2 – 4m2 – 24kr + 16rm + 12km 

= 9k2 – 9k2 + 16r2 – 16r2 + 4m2 – 4m2 – 24kr – 24kr + 16rm + 16rm – 12km + 12km 

= 32rm – 48kr

iii. (7a – 6b + 5c)2 + (7a + 6b – 5c)2 

= [(7a)2 + (- 6b)2 + (5c)2 + 2(7a) (-6b) + 2(-6b) (5c) + 2(7a) (5c)] + [(7a)2 + (6b)2 + (- 5c)2 + 2 (7a) (6b) + 2 (6b) (- 5c) + 2 (7a) (- 5c)] … [(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac] 

= 49a2 + 36b2 + 25c2 – 84ab – 60bc + 70ac + 49a2 + 36b2 + 25c2 + 84ab – 60bc – 70ac 

= 49a2 + 49a2 + 36b2 + 36b2 + 25c2 + 25c2 – 84ab + 84ab – 60bc – 60bc + 70ac – 70ac 

= 98a2 + 72b2 + 50c2 – 120bc
Discussion
7.

Use the formulae to fill proper terms in the following boxes.1. (x + 2y)2 = x2 + ___ + 4y2 2. (2x – 5y)2 = __ – 20xy + __ 3. (101)2 = (100 + 1)2 = ___+ ___ + 12 = ___ 4. (98)2 = (100 – 2)2= 10000 – ___ + ___ = ___ 5. (5m + 3n) (5m – 3n) = ___ – ___ = ___ – __

Answer»

1. (x + 2y)2 = x2 + 4xy + 4y2 

2. (2x – 5y)2 = 4x2 – 20xy + 25y2

3. (101)2 = (100 + 1)2 = 10000+ 200 + 12 = 10201 

4. (98)2 = (100 – 2)2= 10000 – 400 + 4 = 9604

5. (5m + 3n) (5m – 3n) = (5m)2 – (3n)2 = 25m2 – 9n2

  1. (x+2y)2=x2+2×x×2y+(2y)2=x2+4xy+4y2 (Use this formula (a+b)2=a²+2ab+b²)
  2. (2x – 5y)2=4x²-20xy+25y² (Use this formula (a-b)²=a²-2ab+b²)
  3. (101)²=(100+1)²=(100)²+2×100×1+1=10000+200+1=10201(Same formula for it)
  4. (98)²=(100-2)²=10000-2×100×2+4=10004-400=9604 (Same formula use for it)
  5. (5m+3n)(5m-3n)=(5m)²-(3n)²=25m²-9n²

(Use the formula a²-b²=(a+b)(a-b)

Hope you clearly understood this, through the basics formula of alzebra)
Discussion
8.

Fill in the boxes with appropriate terms in the steps of expansion. (2p + 3m + 4n)2= (2p)2 + (3m)2 + __ + 2 × 2p × 3m + 2 × __ × 4n + 2 × 2p × __ = __ + 9m2 + __ + 12pm + __ + __

Answer»

(2p + 3m + 4n)2 

= (2p)2 + (3m)2 + (4n)2 + 2 x 2p x 3m + 2 x 3m x 4n + 2 x 2p x 4n 

= 4p2 + 9m2 + 16n2 + 12pm + 24mn + 16pn
Discussion
9.

Use the above formulae to fill proper terms in the following boxes.1. (x + 2y)2 = x2 + ___ + 4y22. (2x – 5y)2 = __ – 20xy + __3. (101)2 = (100 + 1)2 = ___+ ___ + 12 = ___4. (98)2 = (100 – 2)2 = 10000 – ___ + ___ = ___5. (5m + 3n) (5m – 3n) = ___ – ___ = ___ - ___

Answer»

1. (x + 2y)2 = x2 + 4xy + 4y2 

2. (2x – 5y)2 = 4x2 – 20xy + 25y2 

3. (101)2 = (100 + 1)2 = 10000 + 200 + 12 = 10201 

4. (98)2 = (100 – 2)2 = 10000 – 400 + 4 = 9604 

5. (5m + 3n) (5m – 3n) = (5m)2 – (3n)2 = 25m2 – 9n2
Discussion