untitled

or,

2 2

4

( )

4

(a b) a b

ab







Hence,

2 2

2 2

̧ ̧

¹

·

̈ ̈

©

§ 

̧ ̧ 

¹

·

̈ ̈

©

§ 

a b a b

ab

Remark : Product of any two quantities can be expressed as the difference of two
squares by applying the corollary 6.

Formula 3. a^2 b^2 (ab)(ab)

That is, the difference of the squares of two expressions = sum of two expressions u
difference of two expressions.

Formula 4.(xa)(xb) x^2 (ab)xab

That is, (xa)(xb) x^2 (algebraic sum of aand b)x (the product of a and b)

Extension of Formula for Square
There are three terms in the expression abc. It can be considered the sum of
two terms (ab) and c.

Therefore, by applying formula 1, the square of the expression abc is,

(abc)^2 {(ab)c}^2

=(ab)^2  2 (ab)cc^2

=a^2  2 abb^2  2 ac 2 bcc^2

=a^2 b^2 c^2  2 ab 2 bc 2 ac.

Formula 5.(abc)^2 a^2 b^2 c^2  2 ab 2 bc 2 ac.

Corollary 7. a^2 b^2 c^2 (abc)^2  2 (abbcac)

Corollary 8. 2 (abbcac) (abc)^2 (a^2 b^2 c^2 )

Observe : Applying formula 5, we get,

(i) (abc)^2 {ab(c)}^2

= a^2 b^2 (c)^2  2 ab 2 b(c) 2 a(c)

= a^2 b^2 c^2  2 ab 2 bc 2 ac

(ii)(abc)^2 {a(b)c}^2

= a^2 (b)^2 c^2  2 a(b) 2 (b)c 2 ac

= a^2 b^2 c^2  2 ab 2 bc 2 ac

(iii)(abc)^2 {a(b)(c)}^2

= a^2 (b)^2 (c)^2  2 a(b) 2 (b)(c) 2 a(c)