How do you simplify and find the restrictions for #(2x^2+11x+5)/(3x^2+17x+10)#?

1 Answer
May 21, 2017

Answer:

See a solution process below:

Explanation:

First, factor the numerator and denominator:

#(2x^2 + 11x + 5)/(3x^2 + 17x + 10) => ((2x + 1)(x + 5))/((3x + 2)(x + 5))#

Now, cancel common terms in the numerator and denominator:

#((2x + 1)color(red)(cancel(color(black)((x + 5)))))/((3x + 2)color(red)(cancel(color(black)((x + 5))))) => (2x + 1)/(3x + 2)#

To find the restrictions the denominator cannot be #0# therefore we need to solve for:

#3x^2 + 17x + 10 =#

Or

#(3x + 2)(x + 5) = 0#

Solution 1)

#3x + 2 = 0#

#3x + 2 - color(red)(2) = 0 - color(red)(2)#

#3x + 0 = -2#

#3x = -2#

#(3x)/color(red)(3) = -2/color(red)(3)#

#(color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) = -2/3#

#x = -2/3#

Solution 2)

#x + 5 = 0#

#x + 5 - color(red)(5) = 0 - color(red)(5)#

#x + 0 = -5#

#x = -5#

Therefore, the restrictions are:

#x != -2/3# and #x != -5#