Product of Sum and Difference of Two Terms –
Let us take two terms ‘x’ & ‘y’ and find the product of their sum (x+y) and difference (x-y).
There are some laws which has been proved with logical calculation –
A) (x+y) (x-y) = x(x-y) + y(x-y)
= x² - xy + yx - y²
= x² - y²
Or, x²- y² = (x+y) (x-y)
In other words, the product of the sum and the difference of two terms are equal to the difference of their squares.
This can also be expressed such as –
(first term + second term) X (first term – second term) = (first term)² - (second term)²
There are some example are given below for your better understanding –
Example.1) (m+5)(m-5)
= m(m-5) + 5(m-5)
= m² -5m + 5m – 25
= m² - 5²
Example.2) (2x+3y)(2x-3y)
= 2x(2x-3y) + 3y(2x-3y)
= (2x)²- 6xy + 6xy – (3y)²
= (2x)²- (3y)²
Example.3) (3x-5y)(3x+5y)
= 3x(3x+5y) – 5y(3x+5y)
= (3x)²+ 15xy – 15xy – (5y)²
= (3x)²- (5y)²
Example.4) (2aᶟ+3b²) (2aᶟ- 3b²)
= 2aᶟ(2aᶟ- 3b²) + 3b²(2aᶟ- 3b²)
= (2aᶟ)²- 6aᶟb²+ 6aᶟb²- (3b²)²
= (2aᶟ)²- (3b²)²
5a 3a 5a 3a
Example.5) ( -------- - --------) ( -------- + --------- )
6b 5b 6b 5b
5a 5a 3a 3a 5a 3a
= ------- ( ------- + ------- ) - -------- ( ------- + ------- )
6b 6b 5b 5b 6b 5b
5a 15a² 15a² 3a
= (-------)² + -------- - -------- - (--------)²
6b 30b² 30b² 5b
5a 3a
= (--------)² - (--------)²
6b 5b