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
(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
= ( 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 + -------
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
= ( a²+ ------ )² - 2 X a² X -------
a² a² 1
= 14² - 2 [ we have found here, (a² + -------) = 14 from (2) ]
a²
= 196 – 2 = 194 …………………..(3) (Ans.)