# FIND EXPANSION USING THREE BASIC ALGEBRIAC IDENTITIES

Three basic algebraic identities are

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

a2 - b2 = (a + b) (a - b)

Expand the following using algebraic identities given above.

Problem 1 :

(2x + y)2

Solution :

Here a = 2x and b = y

It looks like (a + b)2

= (2x)2 + 2(2x) y + y2

= 22x2 + 2(2x) y + y2

= 4x2 + 4xy + y2

Problem 2 :

(3a + 2b)2

Solution :

a = 3a and b = 2b

It looks like (a + b)2

= (3a)2 + 2(3a) (2b) + (2b)2

= 32a2 + 12ab + 22b2

= 9a2 + 12ab + 4b2

Problem 3 :

(x2 + 2)2

Solution :

a = x2 and b = 2

It looks like (a + b)2

= (x2)2 + 2(x2) (2) + 22

= x4 + 4x2 + 4

Problem 4 :

(√2 + x)2

Solution :

a = √2 and b = x

It looks like (a + b)2

= (√2)2 + 2(√2) (x) + x2

= 2 + 2√2x + x2

Problem 5 :

(x - 1/x)2

Solution :

a = x and b = 1/x

It looks like (a - b)2

= x2 - 2x (1/x) + (1/x)2

= x2 - 2 + (1/x2)

Problem 6 :

(2 - x2)2

Solution :

a = 2 and b = x2

It looks like (a - b)2

= 22 - 2(2) (x2) + (x2)2

= 4 - 4x2 + x4

Problem 7 :

(8y - 5z)2

Solution :

a = 8y and b = 5z

It looks like (a - b)2

= (8y)2 - 2(8y) (5z) + (5z)2

= 82y2 - 16y (5z) + 52z2

= 64y2 - 80yz + 25z2

Problem 8 :

x2 - 9

Solution :

Here 9 = 32

x2 - 9 = x2 - 32

It looks like a2 - b2

Here a = x and b = 3

x2 - 9 = (x + 3) (x - 3)

Problem 9 :

x2y2 - 1

Solution :

= x2y2 - 1

= (xy)2 - 12

= (xy + 1) (xy - 1)

Problem 10 :

3x2 - 27y2

Solution :

3x2 - 27y2

Factoring 3, we get

= 3(x2 - 9y2)

= 3 (x2 - 32y2)

= 3(x2 - (3y)2)

= 3(x + 3y) (x - 3y)

Problem 11 :

400x2 - 169y2

Solution :

= 400x2 - 169y2

We cannot factorize anything.

400 = 202 and 169 = 132

= ((20x)2 - (13y)2)

= (20x + 13y) (20x - 13y)

Problem 12 :

(a+b)2 - c2

Solution :

Here a = a + b and b = c

Using the algebraic identity a2 - b2, we get

= (a + b + c) (a + b - c)

Problem 13 :

(3x + 1)2 - 9

Solution :

= (3x + 1)2 - 9

= (3x + 1)2 - 32

Here a = 3x + 1 and b = 3

= (3x + 1 + 3) (3x + 1 - 3)

= (3x + 4) (3x - 2)

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