Question

How do you simplify `(4y + 8 )/(y^2 + 7y + 10)`?

Answer 1

`4/(y+5)`

Explanation:

Your starting expression is

`(4y + 8)/(y^2 + 7y + 10)`

Notice that you can factor the numerator like this

`4y + 8 = 4 * (y + 2)`

Now focus on finding a way to factor the denominator. To do that, find the rotts to this quadratic equation by using the quadratic formula

`y^2 + 7y + 10 = 0`

`y_(1,2) = (-7 +- sqrt(7^2 - 4 * 1 * 10))/(2 * 1)`

`y_(1,2) = (-7 +- sqrt(3))/2 = {(y_1 = (-7 - 3)/2 = -5), (y_2 = (-7 + 3)/2 = -2) :}`

This means that the denominator can be factored as

`y^2 + 7y + 10 = [y-(-2)] * [y - (-5)]`

`=(y+2)(y+5)`

Your starting expression can now be written as

`(4 * color(red)(cancel(color(black)((y+2)))))/(color(red)(cancel(color(black)((y+2)))) * (y+5)) = color(green)(4/(y+5))`