Algebraic Expressions and Identities - Exercise 6.6 | Mathematics (Maths) Class 8 PDF Download

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Q u e s t i o n : 9 5
Write the following squares of binomials as trinomials:
i (x + 2)
2
ii (8a + 3b)
2
iii (2m + 1)
2
iv 9a+162
v x+x222
vi x4-y3
vii 3x-13x2
viii xy-yx2
ix 3a2-5b42
x (a
2
b - bc
2
)
2
xi 2a3b+2b3a2
xii (x
2
 - ay)
2
S o l u t i o n :
 We will use the identities a+b2=a2+2ab+b2 and a-b2=a2-2ab+b2 to convert the squares of binomials as trinomials.
(i) x+22=x2+2×x×2+b2=x2+4x+b2
(ii) 8a+3b2=8a2+28a3b+6b2=64a2+48ab+36b2
(iii) 2m+12=2m2+22m1+12=4m2+4m+1
(iv) 9a+162=9a2+29a16+162=81a2+3a+136
(v) x+x222=x2+2xx22+x222=x2+x3+x44
(vi) x4-y32=x42-2x4y3+y32=x216-16xy+y29
(vii) 3x-13x2=3x2-23x13x+13x2=9x2-2+19x2
(viii) xy-yx2=xy2-2xyyx+yx2=x2y2-2+y2x2
(ix) 3a2-5b42=3a22-23a25b4+5b42=9a24-15ab4+25b216
(x) a2b-bc22=a2b2-2a2bbc2+bc22=a4b2-2a2b2c2+b2c4
(xi) 2a3b+2b3a2=2a3b2+22a3b2b3a+2b3a2=4a29b2+89+4b29a2
(xii) x2-ay2=x22-2x2ay+ay2=x4-2x2ay+a2y2
Q u e s t i o n : 9 6
Find the product of the following binomials:
i (2x + y)(2x + y)
ii (a + 2b)(a - 2b)
iii (a
2
 + bc)(a
2 
- bc)
iv 4x5-3y44x5+3y4
v 2x+3y2x-3y
vi (2a
3
 + b
3
)(2a
3
 - b
3
)
vii x4+2x2x4-2x2
viii x3+1x3x3-1x3
S o l u t i o n :
i We will use the identity a+b2=a2+2ab+b2  in the given expression to find the product.
2x+y2x+y=2x+y2=2x2+22xy+y2=4x2+4xy+y2
ii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
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Q u e s t i o n : 9 5
Write the following squares of binomials as trinomials:
i (x + 2)
2
ii (8a + 3b)
2
iii (2m + 1)
2
iv 9a+162
v x+x222
vi x4-y3
vii 3x-13x2
viii xy-yx2
ix 3a2-5b42
x (a
2
b - bc
2
)
2
xi 2a3b+2b3a2
xii (x
2
 - ay)
2
S o l u t i o n :
 We will use the identities a+b2=a2+2ab+b2 and a-b2=a2-2ab+b2 to convert the squares of binomials as trinomials.
(i) x+22=x2+2×x×2+b2=x2+4x+b2
(ii) 8a+3b2=8a2+28a3b+6b2=64a2+48ab+36b2
(iii) 2m+12=2m2+22m1+12=4m2+4m+1
(iv) 9a+162=9a2+29a16+162=81a2+3a+136
(v) x+x222=x2+2xx22+x222=x2+x3+x44
(vi) x4-y32=x42-2x4y3+y32=x216-16xy+y29
(vii) 3x-13x2=3x2-23x13x+13x2=9x2-2+19x2
(viii) xy-yx2=xy2-2xyyx+yx2=x2y2-2+y2x2
(ix) 3a2-5b42=3a22-23a25b4+5b42=9a24-15ab4+25b216
(x) a2b-bc22=a2b2-2a2bbc2+bc22=a4b2-2a2b2c2+b2c4
(xi) 2a3b+2b3a2=2a3b2+22a3b2b3a+2b3a2=4a29b2+89+4b29a2
(xii) x2-ay2=x22-2x2ay+ay2=x4-2x2ay+a2y2
Q u e s t i o n : 9 6
Find the product of the following binomials:
i (2x + y)(2x + y)
ii (a + 2b)(a - 2b)
iii (a
2
 + bc)(a
2 
- bc)
iv 4x5-3y44x5+3y4
v 2x+3y2x-3y
vi (2a
3
 + b
3
)(2a
3
 - b
3
)
vii x4+2x2x4-2x2
viii x3+1x3x3-1x3
S o l u t i o n :
i We will use the identity a+b2=a2+2ab+b2  in the given expression to find the product.
2x+y2x+y=2x+y2=2x2+22xy+y2=4x2+4xy+y2
ii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a+2ba-2b=a2-2b2=a2-4b2
iii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a2+bca2-bc=a22-bc2=a4-b2c2
ivWe will use the identity a+ba-b=a2-b2 in the given expression to find the product.
4x5-3y44x5+3y4=4x52-3y42=16x225-9y216
v We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2x+3y2x-3y=2x2-3y2=4x2-9y2
vi We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2a3+b32a3-b3=2a32-b32=4a6-b6
vii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x4+2x2x4-2x2=x42-2x22=x8-4x4
viii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x3+1x3x3-1x3=x32-1x32=x6-1x6
Q u e s t i o n : 9 7
Using the formula for squaring a binomial, evaluate the following:
i 102
2
ii 99
2
iii 1001
2
iv 999
2
v 703
2
S o l u t i o n :
i Here, we will use the identity a+b2=a2+2ab+b2
1022=100+22=1002+2×100×2+22=10000+400+4=10404
ii Here, we will use the identity a-b2=a2-2ab+b2
992=100-12=1002-2×100×1+12=10000-200+1=9801
iii Here, we will use the identity a+b2=a2+2ab+b2
10012=1000+12=10002+2×1000×1+12=1000000+2000+1=1002001
iv Here, we will use the identity a-b2=a2-2ab+b2
9992=1000-12=10002-2×1000×1+12=1000000-2000+1=998001
v Here, we will use the identity a+b2=a2+2ab+b2
7032=700+32=7002+2×700×3+32=490000+4200+9=494209
Q u e s t i o n : 9 8
Simplify the following using the formula: (a - b)(a + b) = a
2
 - b
2
:
i 82
2
 - 18
2
ii 467
2
 - 33
2
iii 79
2
 - 69
2
iv 197 × 203
v 113 × 87
vi 95 × 105
vii 1.8 × 2.2
viii 9.8 × 10.2
S o l u t i o n :
Here, we will use the identity (a-b)(a+b)=a2 -b2
i Let us consider the following expression:
822-182=82+1882-18=100×64=6400
ii Let us consider the following expression:
4672-332=467+33467-33=500×434=217000
iii Let us consider the following expression:
792-692=79+6979-69=148×10=1480
iv Let us consider the following product:
197×203
? 197+2032=4002=200; therefore, we will write the above product as:
197×203=200-3200+3=2002-32=40000-9=39991
Thus, the answer is 39991.
v Let us consider the following product:
113×87
?113+872=2002=100; therefore, we will write the above product as:
113×87=100+13100-13=1002-132=10000-169=9831
Thus, the answer is 9831.
vi Let us consider the following product:
95×105
?95+1052=2002=100; therefore, we will write the above product as:
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Q u e s t i o n : 9 5
Write the following squares of binomials as trinomials:
i (x + 2)
2
ii (8a + 3b)
2
iii (2m + 1)
2
iv 9a+162
v x+x222
vi x4-y3
vii 3x-13x2
viii xy-yx2
ix 3a2-5b42
x (a
2
b - bc
2
)
2
xi 2a3b+2b3a2
xii (x
2
 - ay)
2
S o l u t i o n :
 We will use the identities a+b2=a2+2ab+b2 and a-b2=a2-2ab+b2 to convert the squares of binomials as trinomials.
(i) x+22=x2+2×x×2+b2=x2+4x+b2
(ii) 8a+3b2=8a2+28a3b+6b2=64a2+48ab+36b2
(iii) 2m+12=2m2+22m1+12=4m2+4m+1
(iv) 9a+162=9a2+29a16+162=81a2+3a+136
(v) x+x222=x2+2xx22+x222=x2+x3+x44
(vi) x4-y32=x42-2x4y3+y32=x216-16xy+y29
(vii) 3x-13x2=3x2-23x13x+13x2=9x2-2+19x2
(viii) xy-yx2=xy2-2xyyx+yx2=x2y2-2+y2x2
(ix) 3a2-5b42=3a22-23a25b4+5b42=9a24-15ab4+25b216
(x) a2b-bc22=a2b2-2a2bbc2+bc22=a4b2-2a2b2c2+b2c4
(xi) 2a3b+2b3a2=2a3b2+22a3b2b3a+2b3a2=4a29b2+89+4b29a2
(xii) x2-ay2=x22-2x2ay+ay2=x4-2x2ay+a2y2
Q u e s t i o n : 9 6
Find the product of the following binomials:
i (2x + y)(2x + y)
ii (a + 2b)(a - 2b)
iii (a
2
 + bc)(a
2 
- bc)
iv 4x5-3y44x5+3y4
v 2x+3y2x-3y
vi (2a
3
 + b
3
)(2a
3
 - b
3
)
vii x4+2x2x4-2x2
viii x3+1x3x3-1x3
S o l u t i o n :
i We will use the identity a+b2=a2+2ab+b2  in the given expression to find the product.
2x+y2x+y=2x+y2=2x2+22xy+y2=4x2+4xy+y2
ii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a+2ba-2b=a2-2b2=a2-4b2
iii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a2+bca2-bc=a22-bc2=a4-b2c2
ivWe will use the identity a+ba-b=a2-b2 in the given expression to find the product.
4x5-3y44x5+3y4=4x52-3y42=16x225-9y216
v We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2x+3y2x-3y=2x2-3y2=4x2-9y2
vi We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2a3+b32a3-b3=2a32-b32=4a6-b6
vii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x4+2x2x4-2x2=x42-2x22=x8-4x4
viii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x3+1x3x3-1x3=x32-1x32=x6-1x6
Q u e s t i o n : 9 7
Using the formula for squaring a binomial, evaluate the following:
i 102
2
ii 99
2
iii 1001
2
iv 999
2
v 703
2
S o l u t i o n :
i Here, we will use the identity a+b2=a2+2ab+b2
1022=100+22=1002+2×100×2+22=10000+400+4=10404
ii Here, we will use the identity a-b2=a2-2ab+b2
992=100-12=1002-2×100×1+12=10000-200+1=9801
iii Here, we will use the identity a+b2=a2+2ab+b2
10012=1000+12=10002+2×1000×1+12=1000000+2000+1=1002001
iv Here, we will use the identity a-b2=a2-2ab+b2
9992=1000-12=10002-2×1000×1+12=1000000-2000+1=998001
v Here, we will use the identity a+b2=a2+2ab+b2
7032=700+32=7002+2×700×3+32=490000+4200+9=494209
Q u e s t i o n : 9 8
Simplify the following using the formula: (a - b)(a + b) = a
2
 - b
2
:
i 82
2
 - 18
2
ii 467
2
 - 33
2
iii 79
2
 - 69
2
iv 197 × 203
v 113 × 87
vi 95 × 105
vii 1.8 × 2.2
viii 9.8 × 10.2
S o l u t i o n :
Here, we will use the identity (a-b)(a+b)=a2 -b2
i Let us consider the following expression:
822-182=82+1882-18=100×64=6400
ii Let us consider the following expression:
4672-332=467+33467-33=500×434=217000
iii Let us consider the following expression:
792-692=79+6979-69=148×10=1480
iv Let us consider the following product:
197×203
? 197+2032=4002=200; therefore, we will write the above product as:
197×203=200-3200+3=2002-32=40000-9=39991
Thus, the answer is 39991.
v Let us consider the following product:
113×87
?113+872=2002=100; therefore, we will write the above product as:
113×87=100+13100-13=1002-132=10000-169=9831
Thus, the answer is 9831.
vi Let us consider the following product:
95×105
?95+1052=2002=100; therefore, we will write the above product as:
95×105=100+5100-5=1002-52=10000-25=9975
Thus, the answer is 9975.
vii Let us consider the following product:
1.8×2.2
?1.8+2.22=42=2; therefore, we will write the above product as:
1.8×2.2=2-0.22+0.2=22-0.22=4-0.04=3.96
Thus, the answer is 3.96.
viii Let us consider the following product:
9.8×10.2
?9.8+10.22=202=10; therefore, we will write the above product as:
9.8×10.2=10-0.210+0.2=102-0.22=100-0.04=99.96
Thus, the answer is 99.96.
Q u e s t i o n : 9 9
Simplify the following using the identities:
i 582-42216
ii 178 × 178 - 22 × 22
iii 198×198-102×10296
iv 1.73 × 1.73 - 0.27 × 0.27
v 8.63×8.63-1.37×1.370.726
S o l u t i o n :
i Let us consider the following expression:
582-42216
Using the identity a+ba-b=a2-b2, we get:
582-42216=58+4258-4216
?582-42216=100×1616 ?582-42216=100
Thus, the answer is 100.
ii Let us consider the following expression:
178×178-22×22
Using the identity a+ba-b=a2-b2, we get:
178×178-22×22=1782-222=178+22178-22=200×156=31200
Thus, the answer is 31200.
iii Let us consider the following expression:
198×198-102×10296=1982-102296
Using the identity a+ba-b=a2-b2, we get:
198×198-102×10296=1982-102296=198+102198-10296
?198×198-102×10296=198+102198-10296 ?198×198-102×10296=300×9696 ?198×198-102×10296=300
Thus, the answer is 300.
iv Let us consider the following expression:
1.73×1.73-0.27×0.27
Using the identity a+ba-b=a2-b2, we get:
1.73×1.73-0.27×0.27=1.732-0.272=1.73+0.271.73-0.27=2×1.46=2.92
Thus, the answer is 2.92.
v Let us consider the following expression:
8.63×8.63-1.37×1.370.726=8.632-1.3720.726
Using the identity a+ba-b=a2-b2, we get:
8.63×8.63-1.37×1.370.726=8.632-1.3720.726=8.63+1.378.63-1.370.726
?8.63×8.63-1.37×1.370.726=8.63+1.378.63-1.370.726 ?8.63×8.63-1.37×1.370.726=8.63+1.378.63-1.370.726 ?8.63×8.63-1.37×1.370.726=10×7.260.726 ?8.63×8.63-1.37×1.370.726=10×7.
?8.63×8.63-1.37×1.370.726=100
Thus, the answer is 100.
Q u e s t i o n : 1 0 0
Find the value of x, if:
i 4x = 52
2
 - 48
2
ii 14x = 47
2
 - 33
2
iii 5x = 50
2
 - 40
2
S o l u t i o n :
i Let us consider the following equation:
4x=522-482
Using the identity a+ba-b=a2-b2, we get:
4x=522-4824x=52+4852-484x=100×4=400
?4x=400
?x=100        Dividing both sides by 4
ii Let us consider the following equation:
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Q u e s t i o n : 9 5
Write the following squares of binomials as trinomials:
i (x + 2)
2
ii (8a + 3b)
2
iii (2m + 1)
2
iv 9a+162
v x+x222
vi x4-y3
vii 3x-13x2
viii xy-yx2
ix 3a2-5b42
x (a
2
b - bc
2
)
2
xi 2a3b+2b3a2
xii (x
2
 - ay)
2
S o l u t i o n :
 We will use the identities a+b2=a2+2ab+b2 and a-b2=a2-2ab+b2 to convert the squares of binomials as trinomials.
(i) x+22=x2+2×x×2+b2=x2+4x+b2
(ii) 8a+3b2=8a2+28a3b+6b2=64a2+48ab+36b2
(iii) 2m+12=2m2+22m1+12=4m2+4m+1
(iv) 9a+162=9a2+29a16+162=81a2+3a+136
(v) x+x222=x2+2xx22+x222=x2+x3+x44
(vi) x4-y32=x42-2x4y3+y32=x216-16xy+y29
(vii) 3x-13x2=3x2-23x13x+13x2=9x2-2+19x2
(viii) xy-yx2=xy2-2xyyx+yx2=x2y2-2+y2x2
(ix) 3a2-5b42=3a22-23a25b4+5b42=9a24-15ab4+25b216
(x) a2b-bc22=a2b2-2a2bbc2+bc22=a4b2-2a2b2c2+b2c4
(xi) 2a3b+2b3a2=2a3b2+22a3b2b3a+2b3a2=4a29b2+89+4b29a2
(xii) x2-ay2=x22-2x2ay+ay2=x4-2x2ay+a2y2
Q u e s t i o n : 9 6
Find the product of the following binomials:
i (2x + y)(2x + y)
ii (a + 2b)(a - 2b)
iii (a
2
 + bc)(a
2 
- bc)
iv 4x5-3y44x5+3y4
v 2x+3y2x-3y
vi (2a
3
 + b
3
)(2a
3
 - b
3
)
vii x4+2x2x4-2x2
viii x3+1x3x3-1x3
S o l u t i o n :
i We will use the identity a+b2=a2+2ab+b2  in the given expression to find the product.
2x+y2x+y=2x+y2=2x2+22xy+y2=4x2+4xy+y2
ii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a+2ba-2b=a2-2b2=a2-4b2
iii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a2+bca2-bc=a22-bc2=a4-b2c2
ivWe will use the identity a+ba-b=a2-b2 in the given expression to find the product.
4x5-3y44x5+3y4=4x52-3y42=16x225-9y216
v We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2x+3y2x-3y=2x2-3y2=4x2-9y2
vi We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2a3+b32a3-b3=2a32-b32=4a6-b6
vii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x4+2x2x4-2x2=x42-2x22=x8-4x4
viii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x3+1x3x3-1x3=x32-1x32=x6-1x6
Q u e s t i o n : 9 7
Using the formula for squaring a binomial, evaluate the following:
i 102
2
ii 99
2
iii 1001
2
iv 999
2
v 703
2
S o l u t i o n :
i Here, we will use the identity a+b2=a2+2ab+b2
1022=100+22=1002+2×100×2+22=10000+400+4=10404
ii Here, we will use the identity a-b2=a2-2ab+b2
992=100-12=1002-2×100×1+12=10000-200+1=9801
iii Here, we will use the identity a+b2=a2+2ab+b2
10012=1000+12=10002+2×1000×1+12=1000000+2000+1=1002001
iv Here, we will use the identity a-b2=a2-2ab+b2
9992=1000-12=10002-2×1000×1+12=1000000-2000+1=998001
v Here, we will use the identity a+b2=a2+2ab+b2
7032=700+32=7002+2×700×3+32=490000+4200+9=494209
Q u e s t i o n : 9 8
Simplify the following using the formula: (a - b)(a + b) = a
2
 - b
2
:
i 82
2
 - 18
2
ii 467
2
 - 33
2
iii 79
2
 - 69
2
iv 197 × 203
v 113 × 87
vi 95 × 105
vii 1.8 × 2.2
viii 9.8 × 10.2
S o l u t i o n :
Here, we will use the identity (a-b)(a+b)=a2 -b2
i Let us consider the following expression:
822-182=82+1882-18=100×64=6400
ii Let us consider the following expression:
4672-332=467+33467-33=500×434=217000
iii Let us consider the following expression:
792-692=79+6979-69=148×10=1480
iv Let us consider the following product:
197×203
? 197+2032=4002=200; therefore, we will write the above product as:
197×203=200-3200+3=2002-32=40000-9=39991
Thus, the answer is 39991.
v Let us consider the following product:
113×87
?113+872=2002=100; therefore, we will write the above product as:
113×87=100+13100-13=1002-132=10000-169=9831
Thus, the answer is 9831.
vi Let us consider the following product:
95×105
?95+1052=2002=100; therefore, we will write the above product as:
95×105=100+5100-5=1002-52=10000-25=9975
Thus, the answer is 9975.
vii Let us consider the following product:
1.8×2.2
?1.8+2.22=42=2; therefore, we will write the above product as:
1.8×2.2=2-0.22+0.2=22-0.22=4-0.04=3.96
Thus, the answer is 3.96.
viii Let us consider the following product:
9.8×10.2
?9.8+10.22=202=10; therefore, we will write the above product as:
9.8×10.2=10-0.210+0.2=102-0.22=100-0.04=99.96
Thus, the answer is 99.96.
Q u e s t i o n : 9 9
Simplify the following using the identities:
i 582-42216
ii 178 × 178 - 22 × 22
iii 198×198-102×10296
iv 1.73 × 1.73 - 0.27 × 0.27
v 8.63×8.63-1.37×1.370.726
S o l u t i o n :
i Let us consider the following expression:
582-42216
Using the identity a+ba-b=a2-b2, we get:
582-42216=58+4258-4216
?582-42216=100×1616 ?582-42216=100
Thus, the answer is 100.
ii Let us consider the following expression:
178×178-22×22
Using the identity a+ba-b=a2-b2, we get:
178×178-22×22=1782-222=178+22178-22=200×156=31200
Thus, the answer is 31200.
iii Let us consider the following expression:
198×198-102×10296=1982-102296
Using the identity a+ba-b=a2-b2, we get:
198×198-102×10296=1982-102296=198+102198-10296
?198×198-102×10296=198+102198-10296 ?198×198-102×10296=300×9696 ?198×198-102×10296=300
Thus, the answer is 300.
iv Let us consider the following expression:
1.73×1.73-0.27×0.27
Using the identity a+ba-b=a2-b2, we get:
1.73×1.73-0.27×0.27=1.732-0.272=1.73+0.271.73-0.27=2×1.46=2.92
Thus, the answer is 2.92.
v Let us consider the following expression:
8.63×8.63-1.37×1.370.726=8.632-1.3720.726
Using the identity a+ba-b=a2-b2, we get:
8.63×8.63-1.37×1.370.726=8.632-1.3720.726=8.63+1.378.63-1.370.726
?8.63×8.63-1.37×1.370.726=8.63+1.378.63-1.370.726 ?8.63×8.63-1.37×1.370.726=8.63+1.378.63-1.370.726 ?8.63×8.63-1.37×1.370.726=10×7.260.726 ?8.63×8.63-1.37×1.370.726=10×7.
?8.63×8.63-1.37×1.370.726=100
Thus, the answer is 100.
Q u e s t i o n : 1 0 0
Find the value of x, if:
i 4x = 52
2
 - 48
2
ii 14x = 47
2
 - 33
2
iii 5x = 50
2
 - 40
2
S o l u t i o n :
i Let us consider the following equation:
4x=522-482
Using the identity a+ba-b=a2-b2, we get:
4x=522-4824x=52+4852-484x=100×4=400
?4x=400
?x=100        Dividing both sides by 4
ii Let us consider the following equation:
14x=472-332
Using the identity a+ba-b=a2-b2, we get:
14x=472-33214x=47+3347-3314x=80×14=1120
?14x=1120
?x=80        Dividing both sides by 14
iii Let us consider the following equation:
5x=502-402
Using the identity a+ba-b=a2-b2, we get:
5x=502-4025x=50+4050-405x=90×10=900
?5x=900
?x=180        Dividing both sides by 5
Q u e s t i o n : 1 0 1
If x+1x=20, find the value of x2+1x2.
S o l u t i o n :
Let us consider the following equation:
x+1x=20
Squaring both sides, we get:
x+1x2=202=400 ?x+1x2=400 ?x2+2×x×1x+1x2=400                   [(a+b)2=a2 +b2 +2ab] ?x2+2+1x2=400
?x2+1x2=398                               Subtracting 2 from both sides
Thus, the answer is 398.
Q u e s t i o n : 1 0 2
If x-1x=3, find the values of x2+1x2 and x4+1x4.
S o l u t i o n :
Let us consider the following equation:
x-1x=3
Squaring both sides, we get:
x-1x2=32=9 ?x-1x2=9 ?x2-2×x×1x+1x2=9 ?x2-2+1x2=9
?x2+1x2=11                               Adding 2 to both sides
Squaring both sides again, we get:
x2+1x22=112=121 ?x2+1x22=121 ?x22+2x21x2+1x22=121 ?x4+2+1x4=121
?x4+1x4=119
Q u e s t i o n : 1 0 3
If x2+1x2=18, find the values of x+1x and x-1x.
S o l u t i o n :
Let us consider the following expression:
x+1x
Squaring the above expression, we get:
x+1x2=x2+2×x×1x+1x2=x2-2+1x2                       [(a+b)2=a2+b2+2ab] ?x+1x2=x2+2+1x2
?x+1x2=20                                                                     ( ? x2+1x2=18)
?x+1x=±20                                                            Taking square root of both sides
Now, let us consider the following expression:
x-1x
Squaring the above expression, we get:
x-1x2=x2-2×x×1x+1x2=x2-2+1x2                     [(a-b)2=a2+b2-2ab] ?x-1x2=x2-2+1x2
?x-1x2=16                                 ( ? x2+1x2=18)
?x-1x=±4                                     Taking square root of both sides
Q u e s t i o n : 1 0 4
If x + y = 4 and xy = 2, find the value of x
2
 + y
2
S o l u t i o n :
We have:
x+y2=x2+2xy+y2 ?x2+y2=x+y2-2xy
?x2+y2=42-2×2                ( ? x+y=4 and xy=2)
?x2+y2=16-4 ?x2+y2=12
Q u e s t i o n : 1 0 5
If x - y = 7 and xy = 9, find the value of x
2
 + y
2
S o l u t i o n :
We have:
x-y2=x2-2xy+y2 ?x2+y2=x-y2+2xy
?x2+y2=72+2×9                     ( ? x-y=7 and xy=9 )
?x2+y2=72+2×9 ?x2+y2=49+18 ?x2+y2=67
Q u e s t i o n : 1 0 6
If 3x + 5y = 11 and xy = 2, find the value of 9x
2 
+ 25y
2
S o l u t i o n :
We have:
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Q u e s t i o n : 9 5
Write the following squares of binomials as trinomials:
i (x + 2)
2
ii (8a + 3b)
2
iii (2m + 1)
2
iv 9a+162
v x+x222
vi x4-y3
vii 3x-13x2
viii xy-yx2
ix 3a2-5b42
x (a
2
b - bc
2
)
2
xi 2a3b+2b3a2
xii (x
2
 - ay)
2
S o l u t i o n :
 We will use the identities a+b2=a2+2ab+b2 and a-b2=a2-2ab+b2 to convert the squares of binomials as trinomials.
(i) x+22=x2+2×x×2+b2=x2+4x+b2
(ii) 8a+3b2=8a2+28a3b+6b2=64a2+48ab+36b2
(iii) 2m+12=2m2+22m1+12=4m2+4m+1
(iv) 9a+162=9a2+29a16+162=81a2+3a+136
(v) x+x222=x2+2xx22+x222=x2+x3+x44
(vi) x4-y32=x42-2x4y3+y32=x216-16xy+y29
(vii) 3x-13x2=3x2-23x13x+13x2=9x2-2+19x2
(viii) xy-yx2=xy2-2xyyx+yx2=x2y2-2+y2x2
(ix) 3a2-5b42=3a22-23a25b4+5b42=9a24-15ab4+25b216
(x) a2b-bc22=a2b2-2a2bbc2+bc22=a4b2-2a2b2c2+b2c4
(xi) 2a3b+2b3a2=2a3b2+22a3b2b3a+2b3a2=4a29b2+89+4b29a2
(xii) x2-ay2=x22-2x2ay+ay2=x4-2x2ay+a2y2
Q u e s t i o n : 9 6
Find the product of the following binomials:
i (2x + y)(2x + y)
ii (a + 2b)(a - 2b)
iii (a
2
 + bc)(a
2 
- bc)
iv 4x5-3y44x5+3y4
v 2x+3y2x-3y
vi (2a
3
 + b
3
)(2a
3
 - b
3
)
vii x4+2x2x4-2x2
viii x3+1x3x3-1x3
S o l u t i o n :
i We will use the identity a+b2=a2+2ab+b2  in the given expression to find the product.
2x+y2x+y=2x+y2=2x2+22xy+y2=4x2+4xy+y2
ii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a+2ba-2b=a2-2b2=a2-4b2
iii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
a2+bca2-bc=a22-bc2=a4-b2c2
ivWe will use the identity a+ba-b=a2-b2 in the given expression to find the product.
4x5-3y44x5+3y4=4x52-3y42=16x225-9y216
v We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2x+3y2x-3y=2x2-3y2=4x2-9y2
vi We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
2a3+b32a3-b3=2a32-b32=4a6-b6
vii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x4+2x2x4-2x2=x42-2x22=x8-4x4
viii We will use the identity a+ba-b=a2-b2 in the given expression to find the product.
x3+1x3x3-1x3=x32-1x32=x6-1x6
Q u e s t i o n : 9 7
Using the formula for squaring a binomial, evaluate the following:
i 102
2
ii 99
2
iii 1001
2
iv 999
2
v 703
2
S o l u t i o n :
i Here, we will use the identity a+b2=a2+2ab+b2
1022=100+22=1002+2×100×2+22=10000+400+4=10404
ii Here, we will use the identity a-b2=a2-2ab+b2
992=100-12=1002-2×100×1+12=10000-200+1=9801
iii Here, we will use the identity a+b2=a2+2ab+b2
10012=1000+12=10002+2×1000×1+12=1000000+2000+1=1002001
iv Here, we will use the identity a-b2=a2-2ab+b2
9992=1000-12=10002-2×1000×1+12=1000000-2000+1=998001
v Here, we will use the identity a+b2=a2+2ab+b2
7032=700+32=7002+2×700×3+32=490000+4200+9=494209
Q u e s t i o n : 9 8
Simplify the following using the formula: (a - b)(a + b) = a
2
 - b
2
:
i 82
2
 - 18
2
ii 467
2
 - 33
2
iii 79
2
 - 69
2
iv 197 × 203
v 113 × 87
vi 95 × 105
vii 1.8 × 2.2
viii 9.8 × 10.2
S o l u t i o n :
Here, we will use the identity (a-b)(a+b)=a2 -b2
i Let us consider the following expression:
822-182=82+1882-18=100×64=6400
ii Let us consider the following expression:
4672-332=467+33467-33=500×434=217000
iii Let us consider the following expression:
792-692=79+6979-69=148×10=1480
iv Let us consider the following product:
197×203
? 197+2032=4002=200; therefore, we will write the above product as:
197×203=200-3200+3=2002-32=40000-9=39991
Thus, the answer is 39991.
v Let us consider the following product:
113×87
?113+872=2002=100; therefore, we will write the above product as:
113×87=100+13100-13=1002-132=10000-169=9831
Thus, the answer is 9831.
vi Let us consider the following product:
95×105
?95+1052=2002=100; therefore, we will write the above product as:
95×105=100+5100-5=1002-52=10000-25=9975
Thus, the answer is 9975.
vii Let us consider the following product:
1.8×2.2
?1.8+2.22=42=2; therefore, we will write the above product as:
1.8×2.2=2-0.22+0.2=22-0.22=4-0.04=3.96
Thus, the answer is 3.96.
viii Let us consider the following product:
9.8×10.2
?9.8+10.22=202=10; therefore, we will write the above product as:
9.8×10.2=10-0.210+0.2=102-0.22=100-0.04=99.96
Thus, the answer is 99.96.
Q u e s t i o n : 9 9
Simplify the following using the identities:
i 582-42216
ii 178 × 178 - 22 × 22
iii 198×198-102×10296
iv 1.73 × 1.73 - 0.27 × 0.27
v 8.63×8.63-1.37×1.370.726
S o l u t i o n :
i Let us consider the following expression:
582-42216
Using the identity a+ba-b=a2-b2, we get:
582-42216=58+4258-4216
?582-42216=100×1616 ?582-42216=100
Thus, the answer is 100.
ii Let us consider the following expression:
178×178-22×22
Using the identity a+ba-b=a2-b2, we get:
178×178-22×22=1782-222=178+22178-22=200×156=31200
Thus, the answer is 31200.
iii Let us consider the following expression:
198×198-102×10296=1982-102296
Using the identity a+ba-b=a2-b2, we get:
198×198-102×10296=1982-102296=198+102198-10296
?198×198-102×10296=198+102198-10296 ?198×198-102×10296=300×9696 ?198×198-102×10296=300
Thus, the answer is 300.
iv Let us consider the following expression:
1.73×1.73-0.27×0.27
Using the identity a+ba-b=a2-b2, we get:
1.73×1.73-0.27×0.27=1.732-0.272=1.73+0.271.73-0.27=2×1.46=2.92
Thus, the answer is 2.92.
v Let us consider the following expression:
8.63×8.63-1.37×1.370.726=8.632-1.3720.726
Using the identity a+ba-b=a2-b2, we get:
8.63×8.63-1.37×1.370.726=8.632-1.3720.726=8.63+1.378.63-1.370.726
?8.63×8.63-1.37×1.370.726=8.63+1.378.63-1.370.726 ?8.63×8.63-1.37×1.370.726=8.63+1.378.63-1.370.726 ?8.63×8.63-1.37×1.370.726=10×7.260.726 ?8.63×8.63-1.37×1.370.726=10×7.
?8.63×8.63-1.37×1.370.726=100
Thus, the answer is 100.
Q u e s t i o n : 1 0 0
Find the value of x, if:
i 4x = 52
2
 - 48
2
ii 14x = 47
2
 - 33
2
iii 5x = 50
2
 - 40
2
S o l u t i o n :
i Let us consider the following equation:
4x=522-482
Using the identity a+ba-b=a2-b2, we get:
4x=522-4824x=52+4852-484x=100×4=400
?4x=400
?x=100        Dividing both sides by 4
ii Let us consider the following equation:
14x=472-332
Using the identity a+ba-b=a2-b2, we get:
14x=472-33214x=47+3347-3314x=80×14=1120
?14x=1120
?x=80        Dividing both sides by 14
iii Let us consider the following equation:
5x=502-402
Using the identity a+ba-b=a2-b2, we get:
5x=502-4025x=50+4050-405x=90×10=900
?5x=900
?x=180        Dividing both sides by 5
Q u e s t i o n : 1 0 1
If x+1x=20, find the value of x2+1x2.
S o l u t i o n :
Let us consider the following equation:
x+1x=20
Squaring both sides, we get:
x+1x2=202=400 ?x+1x2=400 ?x2+2×x×1x+1x2=400                   [(a+b)2=a2 +b2 +2ab] ?x2+2+1x2=400
?x2+1x2=398                               Subtracting 2 from both sides
Thus, the answer is 398.
Q u e s t i o n : 1 0 2
If x-1x=3, find the values of x2+1x2 and x4+1x4.
S o l u t i o n :
Let us consider the following equation:
x-1x=3
Squaring both sides, we get:
x-1x2=32=9 ?x-1x2=9 ?x2-2×x×1x+1x2=9 ?x2-2+1x2=9
?x2+1x2=11                               Adding 2 to both sides
Squaring both sides again, we get:
x2+1x22=112=121 ?x2+1x22=121 ?x22+2x21x2+1x22=121 ?x4+2+1x4=121
?x4+1x4=119
Q u e s t i o n : 1 0 3
If x2+1x2=18, find the values of x+1x and x-1x.
S o l u t i o n :
Let us consider the following expression:
x+1x
Squaring the above expression, we get:
x+1x2=x2+2×x×1x+1x2=x2-2+1x2                       [(a+b)2=a2+b2+2ab] ?x+1x2=x2+2+1x2
?x+1x2=20                                                                     ( ? x2+1x2=18)
?x+1x=±20                                                            Taking square root of both sides
Now, let us consider the following expression:
x-1x
Squaring the above expression, we get:
x-1x2=x2-2×x×1x+1x2=x2-2+1x2                     [(a-b)2=a2+b2-2ab] ?x-1x2=x2-2+1x2
?x-1x2=16                                 ( ? x2+1x2=18)
?x-1x=±4                                     Taking square root of both sides
Q u e s t i o n : 1 0 4
If x + y = 4 and xy = 2, find the value of x
2
 + y
2
S o l u t i o n :
We have:
x+y2=x2+2xy+y2 ?x2+y2=x+y2-2xy
?x2+y2=42-2×2                ( ? x+y=4 and xy=2)
?x2+y2=16-4 ?x2+y2=12
Q u e s t i o n : 1 0 5
If x - y = 7 and xy = 9, find the value of x
2
 + y
2
S o l u t i o n :
We have:
x-y2=x2-2xy+y2 ?x2+y2=x-y2+2xy
?x2+y2=72+2×9                     ( ? x-y=7 and xy=9 )
?x2+y2=72+2×9 ?x2+y2=49+18 ?x2+y2=67
Q u e s t i o n : 1 0 6
If 3x + 5y = 11 and xy = 2, find the value of 9x
2 
+ 25y
2
S o l u t i o n :
We have:
3x+5y2=3x2+23x5y+5y2 ?3x+5y2=9x2+30xy+25y2 ?9x2+25y2=3x+5y2-30xy
?9x2+25y2=112-30×2                          ( ? 3x+5y=11 and xy=2)
?9x2+25y2=121-60 ?9x2+25y2=61
Q u e s t i o n : 1 0 7
Find the values of the following expressions:
i 16x
2
 + 24x + 9, when x=74
ii 64x
2
 + 81y
2
 + 144xy, when x = 11 and y=43
iii 81x
2
 + 16y
2
 - 72xy, when x=23 and y=34
S o l u t i o n :
i Let us consider the following expression:
16x2+24x+9
Now
16x2+24x+9=4x+32                                 (Using identity a+b2=a2+2ab+b2)
?16x2+24x+9=4×74+32            (Substituting x=74) ?16x2+24x+9=7+32 ?16x2+24x+9=102 ?16x2+24x+9=100
ii Let us consider the following expression:
64x2+81y2+144xy
Now
64x2+81y2+144xy=8x+9y2                                (Using identity a+b2=a2+2ab+b2)
?64x2+81y2+144xy=811+9432                  (Substituting x=11 and y=43) ?64x2+81y2+144xy=88+122 ?64x2+81y2+144xy=1002 ?64x2+81y2+144xy=10000
iii Let us consider the following expression:
81x2+16y2-72xy
Now
81x2+16y2-72xy=9x-4y2                                  (Using identity a+b2=a2-2ab+b2)
?81x2+16y2-72xy=923-4342        (Substituting x=23and y=34) ?81x2+16y2-72xy=6-32  ?81x2+16y2-72xy=32 ?81x2+16y2-72xy=9
Q u e s t i o n : 1 0 8
If x+1x=9, find the value of x4+1x4.
S o l u t i o n :
Let us consider the following equation:
x+1x=9
Squaring both sides, we get:
x+1x2=92=81 ?x+1x2=81 ?x2+2×x×1x+1x2=81 ?x2+2+1x2=81
?x2+1x2=79                               Subtracting 2 from both sides
Now, squaring both sides again, we get:
x2+1x22=792=6241 ?x2+1x22=6241 ?x22+2x21x2+1x22=6241 ?x4+2+1x4=6241
?x4+1x4=6239
Q u e s t i o n : 1 0 9
If x+1x=12, find the value of x-1x.
S o l u t i o n :
Let us consider the following equation:
x+1x=12
Squaring both sides, we get:
x+1x2=122=144 ?x+1x2=144 ?x2+2×x×1x+1x2=144                      [ (a+b)2=a2+b2+2ab] ?x2+2+1x2=144
?x2+1x2=142                               Subtracting 2 from both sides
Now
x-1x2=x2-2×x×1x+1x2=x2-2+1x2                         [(a-b)2=a2+b2-2ab] ?x-1x2=x2-2+1x2 ?x-1x2=142-2                                                  ( ? x2+1x2=142) ?x-1x2=140     ?x-1x=±140                   
Q u e s t i o n : 1 1 0
If 2x + 3y = 14 and 2x - 3y = 2, find the value of xy.
[Hint: Use (2x + 3y)
2
 - (2x - 3y)
2
 = 24xy]
S o l u t i o n :
We will use the identity a+ba-b=a2-b2 to obtain the value of xy.
Squaring (2x+3y) and (2x-3y) both and then subtracting them, we get:
2x+3y2-2x-3y2=2x+3y+2x-3y2x+3y-2x-3y=4x×6y=24xy ?2x+3y2-2x-3y2=24xy
?24xy=2x+3y2-2x-3y2 ?24xy=142-22 ?24xy=14+214-2                        ( ? a+ba-b=a2-b2) ?24xy=16×12 ?xy=16×1224                                       (Dividing both sides by 24) ?xy=8
Q u e s t i o n : 1 1 1
If x
2
 + y
2
 = 29 and xy = 2, find the value of
i x + y
ii x - y
iii x
4
 + y
4
S o l u t i o n :
i We have: