Question

How do you divide and simplify `\frac { x ^ { 2} - 9x + 14} { x ^ { 2} + 7x + 12} \div \frac { 3x ^ { 2} - 21x } { 4x ^ { 2} + 16x }`?

Answer 1

See a solution process below:

Explanation:

First, factor each of the terms is the numerator and denominator and rewrite the expression as:

`((x - 7)(x - 2))/((x + 3)(x + 4)) -: (3x(x - 7))/(4x(x + 4))`

Next, rewrite the expression again as:

`(((x - 7)(x - 2))/((x + 3)(x + 4)))/((3x(x - 7))/(4x(x + 4)))`

Then, use this rule of dividing fractions to divide and simplify:

`(color(red)(a)/color(blue)(b))/(color(green)(c)/color(purple)(d)) = (color(red)(a) xx color(purple)(d))/(color(blue)(b) xx color(green)(c))`

`(color(red)((x - 7)(x - 2))/color(blue)((x + 3)(x + 4)))/(color(green)(3x(x - 7))/color(purple)(4x(x + 4))) => (color(red)((x - 7)(x - 2)) xx color(purple)(4x(x + 4)))/(color(blue)((x + 3)(x + 4)) xx color(green)(3x(x - 7)))`

Next, cancel common terms in the numerator and denominator:

`(color(red)(color(black)(cancel(color(red)((x - 7))))(x - 2)) xx color(purple)(4color(black)(cancel(color(purple)(x)))color(black)(cancel(color(purple)((x + 4))))))/(color(blue)((x + 3)color(black)(cancel(color(blue)((x + 4))))) xx color(green)(3color(black)(cancel(color(green)(x)))color(black)(cancel(color(green)((x - 7)))))) => (4(x - 2))/(3(x + 3))`