Expansion Formulae Class 8th Mathematics (new) MHB Solution

Expansion Formulae

Class 8th Mathematics (new) MHB Solution

Class 8^th 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

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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:

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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}

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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

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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