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Expansion Formulae

Class 8th Mathematics (new) MHB Solution
Practice Set 5.1
  1. Expand.(a + 2)(a – 1)
  2. Expand.(m – 4)(m + 6)
  3. Expand.(p + 8)(p – 3)
  4. Expand.(13 + x)(13 – x)
  5. Expand.(3x + 4y)(3x + 5y)
  6. Expand.(9x – 5l)(9x + 3l)
  7. Expand. ( m + {2}/{3} ) ( m - frac {7}/{3} )
  8. Expand. ( x + {1}/{x} ) ( x - frac {1}/{x} )
  9. Expand. ( {1}/{y} + 4 ) ( frac {1}/{y} - 9 )
Practice Set 5.2
  1. Expand(k + 4)3
  2. Expand(7x + 8y)3
  3. Expand(7 + m)3
  4. Expand(52)3
  5. Expand(101)3
  6. Expand ( x + {1}/{x} ) ^{3}
  7. Expand ( 2m + {1}/{5} ) ^{3}
  8. Expand ( {5x}/{y} + frac {y}/{5x} ) ^{3}
Practice Set 5.3
  1. Expand(2m – 5)3
  2. Expand(4 – p)3
  3. Expand(7x – 9y)3
  4. Expand(58)3
  5. Expand(198)3
  6. Expand ( 2p - {1}/{2p} ) ^{3}
  7. Expand ( 1 - {1}/{a} ) ^{3}
  8. Expand ( {x}/{3} - frac {3}/{x} ) ^{3}
  9. Simplify(2a + b)3 –(2a – b)3
  10. Simplify(3r – 2k)3 +(3r + 2k)3
  11. Simplify(4a – 3)3 –(4a + 3)3
  12. Simplify(5x – 7y)3 +(5x + 7y)3
Practice Set 5.4
  1. Expand(2p + q + 5)2
  2. Expand(m + 2n + 3r)2
  3. Expand(3x + 4y – 5p)2
  4. Expand(7m – 3n – 4k)2
  5. Simplify(x – 2y + 3)2 +(x + 2y – 3)2
  6. Simplify(3k – 4r – 2m)2 -(3k + 4r – 2m)2
  7. Simplify(7a – 6b + 5c)2 +(7a + 6b – 5c)2
Practice Set 5.1
Question 1.

Expand.

(a + 2)(a – 1)
Answer:

(a + 2)(a – 1)= a2 + [(2)+(-1)] a + [(2)×(-1)]


{ ∵(x + p)(x + q)= x2 +(p + q)x +(p × q)


Here x = a, p = 2, q = -1 }


= a2 +(2 – 1)a +(-2)


= a2 + 2a – a – 2


= a2 + a – 2
Question 2.

Expand.

(m – 4)(m + 6)
Answer:

(m – 4)(m + 6)= m2 + [(- 4)+(6)] m + [(- 4)× (6)]


{ ∵(x + p)(x + q)= x2 +(p + q)x +(p × q)}


= m2 +(6 – 4)m +(- 24)


= m2 + 6m – 4m – 24


= m2 + 2m – 24
Question 3.

Expand.

(p + 8)(p – 3)
Answer:

(p + 8)(p – 3)= p2 + [(8)+(- 3)] p + [(8)×(- 3)]


{ ∵(x + a)(x + b)= x2 +(a + b)x +(a × b)}


= p2 +(8 – 3)p +(- 24)


=p2 + 8p – 3p – 24


= p2 + 5p – 24
Question 4.

Expand.

(13 + x)(13 – x)
Answer:

(13 + x)(13 – x)=(13)2 –(x)2


{ ∵(a + b)(a – b)=(a)2 –(b)2}


= 169 – x2
Question 5.

Expand.

(3x + 4y)(3x + 5y)
Answer:

(3x + 4y)(3x + 5y)=(3x)2 + [(4y)+(5y)] 3x + [(4y)×(5y)]


{ ∵(x + a)(x + b)= x2 +(a + b)x +(a × b)}


= 9x2 + [(9y)×(3x)] + 20y2


= 9x2 + 27xy + 20y2
Question 6.

Expand.

(9x – 5l)(9x + 3l)
Answer:

(9x – 5l)(9x + 3l)=(9x)2 + [(- 5l)+(3l)] 9x + [(- 5l)×(3l)]


{ ∵(x + a)(x + b)= x2 +(a + b)x +(a × b)}


= 81x2 + [(- 2l)×(9x)] +(- 15l2)


= 81x2 – 18xl – 15l2
Question 7.

Expand.
Answer:


{∵ (x + a)(x + b) = x2 + (a + b)x + (a × b)}
Question 8.

Expand.
Answer:


{∵ (a + b)(a – b) = (a)2 – (b)2}
Question 9.

Expand.
Answer:





Practice Set 5.2
Question 1.

Expand

(k + 4)3
Answer:

(k + 4)3 =(k)3 + [ 3 ×(k)2 ×(4)] + [ 3 ×(k)×(4)2 ] +(4)3


{ ∵(a + b)3 = a3 + 3a2b + 3ab2 + b3


Here a = k, b = 4 }


= k3 +(3 × 4)k2 +(3 × 16)k + 64


= k3 + 12k2 + 48k + 64
Question 2.

Expand

(7x + 8y)3
Answer:

(7x + 8y)3 =(7x)3 + [ 3 ×(7x)2 ×(8y)] + [ 3 ×(7x)×(8y)2 ] + (8y)3


{ ∵ (a + b)3 = a3 + 3a2b + 3ab2 + b3 }


= 343x3 +(3 × 49 × 8)x2y +(3 × 7 × 64)xy2 + 512y3


=343x3 + 1176x2y + 1344xy2 + 512y3
Question 3.

Expand

(7 + m)3
Answer:

(7 + m)3 =(7)3 + [ 3 ×(7)2 ×(m)] + [ 3 ×(7)×(m)2] +(m)3


{ ∵(a + b)3 = a3 + 3a2b + 3ab2 + b3 }


= 343 +(3 × 49)m +(3 × 7)m2 + m3


= 343 + 147m + 21m2 + m3
Question 4.

Expand

(52)3
Answer:

(52)3 =(50 + 2)3


(50 + 2)3 =(50)3 + [ 3 ×(50)2 ×(2)] + [ 3 ×(50)×(2)2] +(2)3


{ ∵ (a + b)3 = a3 + 3a2b + 3ab2 + b3 }


= 125000 +(3 × 2500 × 2)+(3 × 50 × 4) + 8


= 125000 + 15000 + 600 + 8


= 140608
Question 5.

Expand

(101)3
Answer:

(101)3 =(100 + 1)3


(100 + 1)3 =(100)3 + [ 3 ×(100)2 ×(1)] + [ 3 ×(100)×(1)2 ] +(1)3


{ ∵(a + b)3 = a3 + 3a2b + 3ab2 + b3 }


= 1000000 +(3 × 10000 × 1) +(3 × 100 × 1)+ 1


= 1000000 + 30000 + 300 + 1


= 1030301
Question 6.

Expand
Answer:


{∵(a + b)3 = a3 + 3a2b + 3ab2 + b3}
Question 7.

Expand
Answer:


{∵ (a + b)3 = a3 + 3a2b + 3ab2 + b3}
Question 8.

Expand
Answer:


{∵ (a + b)3 = a3 + 3a2b + 3ab2 + b3}
Practice Set 5.3
Question 1.

Expand

(2m – 5)3
Answer:

(2m – 5)3 =(2m)3 – [ 3 ×(2m)2 × 5 ] + [ 3 ×(2m)×(5)2] –(5)3


{ ∵(a – b)3 = a3 – 3a2b + 3ab2 – b3


Here a = 2m, b = - 5 }


= 8m3– [ 3 × 4m2 × 5 ] + [ 3 × 2m × 25] – 125


= 8m3 – 60m2 + 150m – 125
Question 2.

Expand

(4 – p)3
Answer:

(4 – p)3 =(4)3 – [ 3 ×(4)2 × p ] + [ 3 ×(4)×(p)2 ] –(p)3


{ ∵(a – b)3 = a3 – 3a2b + 3ab2 – b3 }


= 64 – [3 × 6 × p ] + [ 3 × 4 × p2 ] – p3


= 64 – 48p + 12p2 – p3
Question 3.

Expand

(7x – 9y)3
Answer:

(7x – 9y)3 =(7x)3 – [ 3 ×(7x)2 × 9y ] + [ 3 ×(7x)×(9y)2 ] –(9y)3


{ ∵(a – b)3 = a3 – 3a2b + 3ab2 – b3 }


= 343x3 – [ 3 × 49x2× 9y ] + [ 3 × 7x × 81y2 ] – 729y3


= 343x3 – 1323x2y + 1701xy2 – 729y3
Question 4.

Expand

(58)3
Answer:

(58)3 =(60 – 2)3


(60 – 2)3 =(60)3 – [ 3 ×(60)2 × 2 ] + [ 3 ×(60)×(2)2 ] –(2)3


{ ∵(a – b )3 = a3 – 3a2b + 3ab2 – b3 }


= 216000 – [ 3 × 3600 × 2 ] + [ 3 × 60 × 4 ] – 8


= 216000 – 21600 + 720 – 8


= 195112
Question 5.

Expand

(198)3
Answer:

(198)3 =(200 – 2)3


(200 – 2)3 =(200)3 – [ 3 ×(200)2 × 2 ] + [ 3 ×(200)×(2)2 ] –(2)3


{ ∵(a – b)3 = a3 – 3a2b + 3ab2 – b3 }


= 8000000 – 240000 + 2400 – 8


= 7762392
Question 6.

Expand
Answer:


{∵ (a – b)3 = a3 – 3a2b + 3ab2 – b3}
Question 7.

Expand
Answer:


{∵ (a – b)3 = a3 – 3a2b + 3ab2 – b3}
Question 8.

Expand
Answer:


{∵ (a – b)3 = a3 – 3a2b + 3ab2 – b3}
Question 9.

Simplify

(2a + b)3 –(2a – b)3
Answer:

(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 and(a – b)3 = a3 – 3a2b + 3ab2 – b3 }


= [ 8a3 +{3 × 4a2× b } +{3 × 2a ×b } + b3 ] – [ 8a3 – { 3 × 4a2× b } +{3 × 2a × b2 } – b3 ]


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


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


= 24a2b + 2b3
Question 10.

Simplify

(3r – 2k)3 +(3r + 2k)3
Answer:

(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 and(a – b)3 = a3 – 3a2b + 3ab2 – b3}


= [ 27r3 –{3 × 9r2 × 2k } + {3 × 3r × 4k2 } – 8k3 ] + [ 27r3 +{3 × 9r2 × 2k } +{3 × 3r ×(4k2 } + 8k3 ]


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


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


= 54r3 + 72rk2
Question 11.

Simplify

(4a – 3)3 –(4a + 3)3
Answer:

(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 and (a – b)3 = a3 –3a2b + 3ab2 – b3 }


= [ 64a3-{3 × 16a2 × 3 } +{3 × 4a × 9} - 27 ] – [ 64a3 +{3 × 16a2 × 3 } +{3 × 4a × 9} + 27 ]


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


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


= - 288a2 - 54
Question 12.

Simplify

(5x – 7y)3 +(5x + 7y)3
Answer:

(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 and(a – b)3 = a3 – 3a2b + 3ab2 – b3}


= [ 125x3 –{3 × 25x2 × 7y } +{3 × 5x × 49y2 } – 343y3 ] + [ 125x3 +{3 × 25x2 × 7y } +{3 × 5x × 49y2 } + 343y3 ]


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


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


= 250x3 + 1470xy2
Practice Set 5.4
Question 1.

Expand

(2p + q + 5)2
Answer:

(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


Here a = 2p, b = q, c = 5 }


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


= 4p2 + q2 + 25 + 4pq + 10q + 20p
Question 2.

Expand

(m + 2n + 3r)2
Answer:

(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 ]


= m2 + 4n2 + 9r2 + 4mn + 12nr + 6mr
Question 3.

Expand

(3x + 4y – 5p)2
Answer:

(3x + 4y – 5p)2 =(3x)2 +(4y)2 +(- 5p)2 + [ 2 ×(3x) ×(4y)] + [ 2 ×(4y)×(- 5p)] + [ 2 ×(3x)×(- 5p)]


{ ∵(a + b +c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac }


= 9x2 + 16y2 + 25p2 + [ 24xy ] + [ – 40yp ] + [– 30xp ]


= 9x2 + 16y2 + 25p2 + 24xy – 40yp – 30xp
Question 4.

Expand

(7m – 3n – 4k)2
Answer:

(7m – 3n – 4k)2 =(7m)2 +(- 3n)2 +(- 4k)2 + [ 2 ×(7m)×(-3n)] + [ 2 ×(-3n)×(-4k)] + [ 2 × (7m)×(-4k)]


{ ∵(a + b +c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac }


=49m2 + 9n2 + 16k2 + [ – 42mn ] + [ 24nk ] + [ – 56mk ]


= 49m2 + 9n2 + 16k2 – 42mn + 24nk – 56mk
Question 5.

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 } ]


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


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


= 2x2 + 8y2 + 18 – 24y
Question 6.

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


= - 48kr + 32rm


= 32rm – 48kr
Question 7.

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


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


= 98a2 + 72b2 + 50c2 – 120bc