# CLASS-9ALGEBRA - FORMULAE OF EXPANSION

- ALGEBRA

FORMULAE OF EXPANSIONS -

1)  (a + b)² = a² + 2ab + b²

2)  (a – b)² = a² - 2ab + b²

3)  (a + b)² + (a – b)² =  2 (a² + b²)

4)  (a + b)² - (a – b)² =  4ab

5)  (a + b) (a – b) =  a² - b²

1                    1

6)  (a + ----- )²  =  a² + ------ + 2

a                    a²

1                     1

7)    (a - ------ )²  =  a² + ------ - 2

a                     a²

1              1                     1

8)   (a + ------ ) (a - ------ )  =   a² - ------

a              a                     a²

1                    1                         1

9)   (a + ------ )²  +  (a - ------ )²  =   2 (a² + ------ )

a                    a                         a²

1                    1

10)   (a + ------ )² -  (a - ------- )²  =   4

a                    a

11)   ( a + b + c )² =  a² + b² + c² + 2 (ab + bc + ca)

12)   (a)  (x + a) (x + b) = x² + (a + b) x + ab

(b)  (x + a) (x - b) = x² + (a – b) x – ab

(c)  (x – a) (x + b ) = x² - (a – b) x – ab

(d)  (x – a) (x – b) = x² - (a + b) x + ab

Some more product

13) (x + a) (x + b) =  (x + a) x + (x + a) b

=  x² + ax + bx + ab

=  x² + (a + b)x + ab

So, (x + a) (x + b)

= x² + (algebraic sum of 2nd terms) x + (product of 2nd terms)

14) (x – a) (x – b)  =  (x – a) x – (x – a) b

=   x² - ax – bx + ab

=   x² - (a + b)x + ab

So, (x - a) (x - b)

= x² - (algebraic sum of 2nd terms) x + (product of 2nd terms)

15)  (x + a) (x – b)  =  (x + a) x – (x + a) b

=  x² + ax – bx – ab

=  x² + (a – b)x – ab

So, (x + a) (x - b) = x² + (algebraic subtraction of 2nd terms) x - (product of 2nd terms)

16)  (x – a) (x + b)(x – a) x + (x – a) b

=  x² - ax + bx – ab

=   x² - (a – b)x - ab

So, (x - a) (x + b)

= x² - (algebraic subtraction of 2nd terms) x - (product of 2nd terms)

To find the product (ax + by) (cx + dy)

Method –

Step.1) Multiply the first term

Step.2) Find the inner and outer products and take their algebraic sums

Step.3) Multiply the last terms,

Thus, we have –

(ax + by) (cx + dy) =  (ax) (cx + dy) + (by) (cx + dy)

=  (ax) (cx) + (ax) (dy) + (by) (cx) + (by) (dy)

=  (ax) (cx) + {(by) (cx) + (ax) (dy)} + (by) (dy)

Here, (by) (cx) is inner product and (ax) (dy) is the outer product.