Reduce this algebraic fraction.

Question

(4x^2-y^2)              ( 8x^2+10xy+3y^2)

----------------    *        ----------------------------

(4x^2-9xy-9y^2)    (2x^2-5xy-3y^2)

Answer 1

Given fraaction is  [(4x^2 - y^2) / (4x^2 - 9xy - 9y^2)]  * [(8x^2 + 10xy + 3y^2) / (2x^2 - 5xy - 3y^2)]

By factoring all the terms we get,

4x^2 - y^2 = (2x - y)(2x + y)   [ Since a^2 - b^2 = (a + b) (a - b) ]

4x^2 - 9xy - 9y^2 = 4x^2 - 12xy + 3xy - 9y^2

                          = 4x(x - 3y) +3y(x - 3y)

                          = (4x + 3y) (x - 3y)

8x^2 + 10xy + 3y^2 = 8x^2 + 4xy + 6xy + 3y^2

                              = 4x(2x + y) + 3y(2x + y)

                              = (4x + 3y)(2x + y)

2x^2 - 5xy - 3y^2 = 2x^2 + xy - 6xy -3y^2

                          = x(2x + y) - 3y(2x + y)

                          = (2x + y)(x - 3y)

Therefore [(4x^2 - y^2) / (4x^2 - 9xy - 9y^2)]  * [(8x^2 + 10xy + 3y^2) / (2x^2 - 5xy - 3y^2)]

                 = [(2x - y)(2x + y) / (4x - 3y) (x - 3y)]  * [(4x + 3y)(2x + y) /(2x + y)(x - 3y)]

                 = (2x - y)(2x + y)(4x + 3y)(2x + y) / (4x + 3y) (x - 3y)(2x + y)(x - 3y)

                 =  (2x - y)(2x + y) / (x - 3y)^2

                 = (4x^2 - y^2) / (x^2 - 6xy + 9y^2)