Difference of Two Square –
The difference of two squares can always be factorized by the formula - a²- b²= (a+b)(a-b)
Example.1) Factorize 25a² - 16b²
Ans.) As we know that, 25a² = (5a)² and 16b² = (4b)²
So, 25a²- 16b² = (5a)²- (4b)² = (5a+4b)(5a-4b) (Ans.)
Example.2) Factorize 3a²- 27b²
Ans.) The HCF of the numerical coefficients in the two terms = 3, so, we take 3 out as the common factor
As we know, a² = (a)² and 9b² = (3b)²
So, 3a²- 27b² = 3(a²- 9b²) = 3{(a)² - (3b)²}
= 3 (a+3b)(a-3b) (Ans.)
Example.3) Factorize 9(a+b)² - 25(x-y)²
Ans.) 9(a+b)²- 25(x-y)² = {3(a+b)}²- {5(x-y)}²
= {3(a+b) + 5(x-y)} {3(a+b) - 5(x-y)}
= (3a+3b+5x-5y) (3a+3b-5x+5y) (Ans.)
Example.4) Factorize 16x²- y²+ 10y – 25
Ans.) 16x²- y² + 10y – 25
= 16x² - (y² - 10y + 25)
= 16x² - {y² - 2.y.5 + (5)²}
= 16x² - (y-5)² = (4x)² - (y-5)²
= (4x+y-5)(4x-y+5) (Ans.)
Example.5) Factorize 2x⁴ - 32y⁴
Ans.) 2x⁴ - 32y⁴
= 2 (x⁴ - 16y⁴) = 2{(x²)²- (4y²)²}
= 2{(x² + 4y²) (x²- 4y²)}
= 2(x²+4y²) {(x)² - (2y)²}
= 2(x²+4y²) (x+2y)(x-2y) (Ans.)