# CLASS-8FINDING EXPANSION OF BINOMIAL

Find the Square of each of the following Binomials –

1

Example.1)    2a + --------

2a

Ans.)   we will use (a+b)² =  a² + 2ab + b²

1                              1          1                        1

(2a + -----)² = (2a)²+ 2 X 2a X ------ + (-----)² =  4a²+ 2 + ------

2a                            2a         2a                      4a²

Example.2) Use special expansion to find a) 202² and b) (59.9)²

Ans.) a)  202² =  (200 + 2)²

=  200²+ 2 X 200 X 2 + 2²

[Applying the formula (a+b)²= a²+2ab+b²]

=  40000 + 800 + 4

=   40804          (Ans.)

b)  (59.9)² =  (60 – 0.1)²

=  60² - 2 X 60 X (0.1) + (0.1)²

[Applying the formula (a-b)² = a²-2ab+b²]

=  3600 – 12 + 0.01

=  3588.01                 (Ans.)

Example.3) If, x²+ y² = 65 and xy = 8, find the values of a) x + y  and  b) x – y

Ans.)   as we all know that, x² + y² = (x + y)² - 2xy, here is the given conditions  are x² + y² = 50 and xy = 8.

As per the law, x²+ y² = (x+y)² - 2xy

So, 65 = (x+y)²- (2X8)

[ to put the value of x²+ y² and xy, i.e. 65 & 8 respectively

(x+y)² =  65 + 16

(x+y)² = 81

(x+y) = ± 9 ……………………….(a)

Now we have to find out the value of (x-y) where x² + y² = 65 and xy = 8

So,  (x – y)² =  x² + y² - 2xy

(x – y)² =  65 – (2X8)

[ to put the value of  x² + y² and xy, i.e. 65 & 8 respectively

(x – y)² =  49

x – y  = ± 7………………..(b)

1                                              1

Example.4) If, a + ----- =  4, then find the value of (1) a - -----,

a                                              a

1                  1

(2) a²+ ------,  (3) a⁴+ ------

a²                 a⁴

1

Ans.) Here is the given condition is a + ------ = 4, we have to find out

a

1

the value of a - -------

a

1

so, as the formula, we know that => ( a - -------)²

a

1                      1

=  ( a + ------)² - 4 X a X -------

a                      a

1

so,  ( a - -------)² =  4² - 4  = 16 – 4  =   12

a

1

so,  ( a - -------)  =  √12  =  2√3………………………(1)        (Ans.)

a

1

Now, we have to find out the value of  a² + -------

1

as per the rules, we know that => a²+ --------

a²

1                       1

=  (a + -------)² - 2 X a X --------

a                       a

1

or,  a² + ------- =   4² - 2  =  16 – 2  =  14 …………………..(2)    (Ans.)

a²

1

Now, we have to find out value of  a⁴ + -------

a⁴

1

As, we all know  (a⁴ + -------)

a⁴

1                        1

=  ( a²+ ------ )² - 2 X a² X -------

a²                                                                                                                                                                                                        1

=  14² - 2    [ we have found here, (a² + -------) = 14 from (2) ]

= 196 – 2 = 194 …………………..(3)           (Ans.)