Question

How do you simplify `\frac { 4x ^ { 2} - 9} { 2x ^ { 2} + x - 6} \div \frac { 8x + 12} { x ^ { 2} - 2x + 8}`?

Answer 1

`(x^2-2x+8)/(4(x+2))`

Explanation:

A first step with algebraic fractions is to factorise:

`(color(blue)(4x^2-9))/(color(red)(2x^2+x-6)) div color(purple)((8x+12))/color(forestgreen)(x^2-2x+8)`

`(color(blue)((2x+3)(2x-3)))/(color(red)((2x-3)(x+2))) xx color(forestgreen)(x^2-2x+8)/color(purple)(4(2x+3))" "larr` multiply and flip

`(cancel((2x+3))cancel((2x-3)))/(cancel((2x-3))(x+2)) xx (x^2-2x+8)/(4cancel((2x+3)))" "larr` cancel

`=(x^2-2x+8)/(4(x+2))`