Algebraic Identities with examples Class 8
Algebraic identities
Formula 1 : ( x + a ) ( x + b ) = x2 + ( a + b ) x + ab
Formula 2 : ( x + a ) ( x - b ) = x2 + ( a - b ) x - ab
Formula 3 : ( x - a ) ( x + b ) = x2 + ( b - a ) x - ab
Formula 4 : ( x - a ) ( x - b ) = x2 - ( a + b ) x + ab
Exercise 1 : Find the following Products :
Q 1.
i) ( x + 5 ) ( x + 7 ) = x2 + ( 5 + 7 ) x + 5*7
= x2 + 12x + 35
ii) ( b + 2 ) ( b + 9 ) = b2 + ( 2 + 9 ) b + 2*9
= b2 + 11b + 18
iii) ( c + 2 ) ( c + 3/5 ) = c2 + ( 2 + 3/5 ) c + 2*3/5
= c2 + 13/5 c + 6/5
iv) ( t + 4/3 ) ( t+ 1/3 ) = t2 + ( 4/3 + 1/3 ) t + 4/3*1/3
= t2 + 5/3 t + 4/9
Q 2 .
i) ( y + 8 ) ( y - 4 ) = y2 + ( 8 - 4 ) y - 8 * 4
= y2 + 4 y - 32
ii) ( z + 6 ) ( z - 11 ) = z2 + ( 6 - 11 )z - 6 * 11
= z2 - 5z - 66
iii) ( c - 5 ) ( c + 1 ) = c2 + (-5 + 1 )c - 5 * 1
= c2 - 4c - 5
iv) ( b - 13 ) ( b + 10 ) = b2 + (-13 + 10 )b - 13 * 10
= b2 - 3b - 130
Q 3 .
i) ( x - 3 ) ( x - 6 ) = x2 - 3x - 6 x + (-3* -6)
= b2 - 14b + 48
iv) ( a - 3/5 ) ( a - 1/3 ) = a2 - 3/5a - 1/3a + (- 3/5 * -1/3 )
= a2 - 14/15a + 3/15
= x2 + 2/2x - 3/4
= x2 + x - 3/4
ii) ( p - 3 ) ( p + 1/2) = p2 + (-3 + 1/2 )p - 3 * 1/2
= p2 - 5/2p - 3/2
Q 5.
i ) ( 4p + 3 ) ( 4p + 7 ) = 16p2 + ( 3 + 7 ) 4p + 3 * 7
= 16p2 + ( 3 + 7 ) 4p + 3 * 7
= 16p2 + 40p + 21
ii) ( 9c + 4 ) ( 9c - 2 ) = 81c2 + ( 4 - 2 )9c - 4 * 2
= 81c2 + 18c - 8
iii) ( 3a - 8 ) ( 3a + 2 ) = 9a2 + (-8 + 2 )3a - 8 * 2
= 9a2 - 18a - 16
iv) ( 5x - 2 ) ( 5x - 7 ) = 25x2 + (- 2 - 7) 5x + (-2 * -7)
i) ( ab - 2 ) ( ab + 4) = a2b2 + ( -2 + 4 )ab + ( -2 * 4)
= a2b2+ 2ab - 8
ii) ( 2 + xy ) ( 3 - xy) = 2 * 3 - 2xy + 3xy - x2y2
= 6 + xy - x2y2
Q 8.
i ) ( 3x + 4y) ( 4x + 3y) = 3x * 4x + 3x * 3y + 4y*4x + 4y * 3y
= 12x2 + 9xy + 16xy + 12y2
= 12x2 + 25xy + 12y2
ii) ( 4a - 5b) ( 3a + 2b) = 4a * 3a + 4a * 2b - 5b * 3a - 5b * 2b
= 12a2 + 8ab - 15ab - 10b2
= 12a2 - 7ab - 10b2
iii) ( 2y + z ) ( 7z - 3y) =2y *7z - 2y * 3y + z * 7z - z * 3y
= 14yz - 6y2 + 7z2 - 3yz
= 11yz - 6y2 + 7z2
iv) ( 2m - 4n ) ( 4m - 3n) = 2m * 4m - 2m * 3n - 4n * 4m + 4n * 3n
= 8m2 -6mn - 16mn + 12n2
= 8m2 - 22mn + 12n2
Q 9 .
i) ( 2pq + 0.1mn ) ( 0.2pq + 3mn )
ii) ( 4m/p - o.2n/q) (3m/p + 0.5n/q)
iii) (3/4 x - 2pq ) ( 3x - 4/5 pq)
Algebraic Identities Class 8
Product of Sum and Difference of two terms
Formula : ( a + b ) ( a - b ) = ( a2 - b2)
Exercise :
Q 1)
i) ( y + 9 ) ( y - 9 ) = ( y2 - 92) = y2 - 81
ii) ( 4 + b ) ( 4 - b ) = ( 42 - b2) = 16 - b2
iii) ( z + 1/2 ) ( z - 1/2 ) = ( z2 - 1/22) = z2 - 1/4
iv) ( a - 2/3 ) ( a + 2/3 ) = ( a2 - 2/32) = a2 - 4/9
Q 2)
i) ( 3x - 5 ) ( 3x + 5 ) = ( 3x2 - 52) = 9x2 - 25
ii) ( 2 + 7x ) ( 2 - 7x ) = ( 22 - 7x2) = 4 - 49x2
iii) ( a/2 + 3 ) ( a/2 - 3 ) = ( a/22 - 32) = a2/4 - 9
iv) ( 4x + 3y ) ( 4x - 3y ) = ( 4x)2 - (3y)2 = 16x2 - 9y2
Q 3)
i) ( a/3 - b/4 ) ( a/3 + b/4 ) = (( a/3)2 - (b/4)2) = a2/9 - b2/16
i) ( t/2 - 1/3 ) ( t/2 + 1/3 ) = (( t/2)2 - (1/3)2) = t2/4 - 1/9
Q 4)
i) ( 2/x + 3/y ) ( 2/x + 3/y ) = (( 2/x)2 - (3/y)2) = 4/x2 - 9/y2
ii) ( 1/a - 1/b ) ( 1/a + 1/b ) = (( 1/a)2 - (1/b)2) = 1/a2 - 1/b2
iii) ( 1/3x + 2/5y ) ( 1/3x - 2/5y ) = (( 1/3x)2 - (2/5y)2) = 1/9x2 - 4/25y2
iv) ( 1.1x - 0.3y ) ( 1.1x + 0.3y ) = ( 1.1x)2 - (0.3y)2 = 1.21x2 - 0.09y2
Q 5)
i) ( a2 + 2b2) ( a2 - 2b2) = ( a2 )2 - ( 2b2 )2 = a4- 4b4
ii) ( 6x2 - 7y2) ( 6x2 + 7y2) = ( 6x2 )2 - ( 7y2 )2 = 36x4- 49y4
iii) ( 4x2 + 2yz ) ( 2x2 - yz) = 2( 2x2 + yz ) ( 2x2 - yz)
= 2 ( 2x2 )2 - ( yz )2
= 2 (4x4- y2 z2) = 8x4- 2y2 z2
iv) ( ab - 3/2cd ) ( 2ab + 3cd) = ( ab - 3/2cd ) 2( ab + 3/2cd)
= 2 ( ab )2 - ( 3/2cd )2
= 2 (a2 b2- 9/4c2 d2) = 2a2 b2- 9/2c2 d2
Q 6)
i) (2x +3) (2x -3) ( 4x2 + 9)
ii) (x + 2y) (x - 2y) ( x2 + 4y2)
iii) (a + bc) (a - bc) ( a2 + b2 c2)
iv) (2/5 + x) (2/5 - x) ( 4/25 + x2 )
Q 7)
i) 88 * 112 = ( 100 -12) ( 100 + 12) = 100 2- 12 2 = 10000 - 144 = 9856
ii) 153 * 167 = ( 160 - 7) ( 160 + 7) = 160 2- 7 2 = 25600 - 49 = 25551
ii) 10.8 * 9.2 = ( 10 + 0.8) ( 10 - 0.8) = 10 2- 0.8 2 = 100 - 0.64 = 99.36
Algebraic Identities
Squares of Binomials
Formula : ( a + b )2 = a² + b² + 2ab
Formula : ( a - b )2 = a² + b² - 2ab
Exercise :
Q 1 )
i) ( x + 3 )2 = x² + 3² + 2* x * 3
= x² + 9 + 6x
ii) ( 2a + 7 )2 = (2a)² + 7² + 2* 2a * 7
= 4a² + 49 + 28a
iii) ( 8 + 3p )2 = (8)² + (3p)² + 2* 8 * 3p
= 64 + 9p² + 48p
iv) ( √3 x + 2)2 = (√3x)² + (2)² + 2* √3 x * 2
= 3x² + 4 + 4√3x
v) ( 4 + √5 y )2 = (4)² + (√5y)² + 2* 4 * √5 y
= 16 + 5y² + 8√5y
vi) ( 6x + 11y )2 = (6x)² + (11y)² + 2* 6x * 11y
= 36x² + 121y + 132xy
vii) ( x/2 + y/3 )2 = (x/2)² + (y/3)² + 2* x/2 * y/3
= x²/4 + y²/9 + 2xy/6
= x²/4 + y²/9 + xy/3
viii) ( 3a/5 + 5b/3 )2 = (3a/5)² + (5b/3)² + 2* 3a/5 * 5b/3
= 9a²/25 + 25b²/9 + 30ab/15
= 9a²/25 + 25b²/9 + 2ab
Q 2 Expand :
i) ( x - 9 )2 = x² + 9² - 2* x * 9
= x² + 81 -18x
ii) ( 6 - y )2 = (6)² + (y)² - 2* 6 * y
= 36 + y² - 12p
iii) ( 3a - 2 )2 = (3a)² + 2² - 2* 3a * 2
= 9a² + 4 - 12a
iv) ( 8y - 5z )2 = (8y)² + (5z)² + 2* 8y * 5z
= 64y² + 25z² - 80yz
v) ( x/2 - y/2 )2 = (x/2)² + (y/2)² - 2* x/2 * y/2
= x²/4 + y²/4 - 2xy/4
= x²/4 + y²/4 - xy/2
vi) ( 2a - 5/2 )2 = (2a)² + (5/2)² - 2* 2a * 5/2
= 4a² + 25/4 - 10a
vii) ( 2/a - 3/b )2 = (2/a)² + (3/b)² - 2* 2/a * 3/b
= 4/a² + 9/b² - 12/ab
viii) ( 3x - 1/3x )2 = (3x)² + (1/3x)² - 2* 3x * 1/3x
= 9x² + 1/9x² - 2
3. Using Special expansions of Algebraic Identities, find the value of :
i) (53)2 = ( 50 + 3)2
= 502 + 32 + 2*50*3
= 2500 + 9 + 300
= 2809
ii) (84)2 = ( 80 + 4)2
= 802 + 42 + 2*80*4
= 6400 + 16 + 640
= 7056
iii) (1011)2 = ( 1000 + 11)2
= 10002 + 112 + 2*1000*11
= 1000000 + 121 + 22000
= 1022121
iv) (10.9)2 = ( 10 + 0.9)2
= 102 + 0.92 + 2*10*0.9
= 100 + 0.81 + 18
= 118.81
4. Using special expansion of Algebraic Identities, find the value of :
i) (67)2 = ( 70 - 3)2
= 702 + 32 - 2*70*3
= 4900 + 9 - 420
= 4489
ii) (795)2 = ( 800 - 5)2
= 8002 + 52 - 2*800*5
= 640000 + 25 - 8000
= 632025
iii) (988)2 = ( 1000 - 12)2
= 10002 + 122 - 2*1000*12
= 1000000 + 144 - 24000
= 976144
iv) (9.2)2 = ( 10 - 0.8)2
= 102 + 0.82 - 2*10*0.8
= 100 + 0.64 - 16
= 84.64
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