Question

How do you simplify `\frac { 4x ^ { 2} - 100} { 3x ^ { 2} - 9x - 30}`?

Answer 1

See a solution process below:

Explanation:

First, factor a common term out of the numerator and denominator:

`(4(x^2 - 25))/(3(x^2 - 3x - 10))`

Next, factor the denominator as:

`(4(x^2 - 25))/(3(x - 5)(x + 2))`

Next, use this rule for quadratics to factor the numerator:

`(color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y)) = color(red)(x)^2 - color(blue)(y)^2`

`(4(x + 5)(x - 5))/(3(x - 5)(x + 2))`

Then, cancel common terms in the numerator and denominator:

`(4(x + 5)color(red)(cancel(color(black)((x - 5)))))/(3color(red)(cancel(color(black)((x - 5)))(x + 2)) =>`

`(4(x + 5))/(3(x + 2))`

Or

`(4x + 20)/(3x + 6)`

Answer 2

`(4(x+5))/(3(x+2))`

Explanation:

`"factorise numerator/denominator"`

`rArr(4(x^2-25))/(3(x^2-3x-10))larrcolor(blue)"common factors"`

`"numerator is a "color(blue)"difference of squares"`

`=(4(x-5)(x+5))/(3(x-5)(x+2))`

`=(4cancel((x-5))(x+5))/(3cancel((x-5))(x+2))`

`=(4(x+5))/(3(x+2))to(x!=-2)`