How do you simplify #(6x^2-54x+84)/(8x^2-40x+48) div (x^2+x-56)/(x^2+12x+32)#?

1 Answer
Jul 30, 2015

Answer:

#E = 3/4 * (x+4)/(x-3)#

Explanation:

Start by writing down your starting expression

#E = (6x^2 - 54x + 84)/(8x^2 - 40x + 48) * (x^2 + 12x + 32)/(x^2 + x - 56)#

For the first fraction, you can factor the numerator by #6# and the denominator by #8# to get

#E = (6 * (x^2 - 9x + 14))/(8 * (x^2 - 5x + 6)) * (x^2 + 12x + 32)/(x^2 + x - 56)#

All these quadratics can be easily factored by using the sum/product technique to get

#x^2 - 9x + 14 = x^2 - 2x - 7x + 14 = (x-2)(x-7)#

#x^2 - 5x + 6 = x^2 - 2x - 3x + 6 = (x-2)(x-3)#

#x^2 + 12x + 32 = x^2 + 4x + 8x + 32 = (x+ 8)(x + 4)#

#x^2 + x - 56 = x^2 + 8x - 7x + 56 = (x+8)(x-7)#

The expression will thus be equal to

#E = (6 * color(red)(cancel(color(black)((x-2)))) * color(green)(cancel(color(black)((x-7)))))/(8 * color(red)(cancel(color(black)((x-2)))) * (x-3)) * (color(blue)(cancel(color(black)((x+8)))) * (x+4))/(color(blue)(cancel(color(black)((x+8)))) * color(green)(cancel(color(black)((x-7)))))#

#E = color(green)(3/4 * (x+4)/(x-3)#