Question

What is the simplified form of the quantity of x plus 9, all over the quantity of 2x plus 3 + the quantity of x plus 4, all over the quantity of x plus 2?

Answer 1

`(3x^2+22x+30)/((2x+3)*(x+2))`

Explanation:

We have been given an expression
`(x+9)/(2x+3)+(x+4)/(x+2)`. We are asked to simplify our given expression.

First of all, we will make a common denominator as shown below:

`((x+9)*(x+2))/((2x+3)*(x+2))+((x+4)*(2x+3))/((x+2)*(2x+3))`

Now, we will use distributive property to solve our expression as:

`(x(x+2)+9(x+2))/((2x+3)*(x+2))+(x(2x+3)+4(2x+3))/((x+2)*(2x+3))`

`(x^2+2x+9x+18)/((2x+3)*(x+2))+(2x^2+3x+8x+12)/((x+2)*(2x+3))`

`(x^2+11x+18)/((2x+3)*(x+2))+(2x^2+11x+12)/((x+2)*(2x+3))`

Now, we will add numerators as:

`(x^2+11x+18+2x^2+11x+12)/((2x+3)*(x+2))`

Combine like terms:

`(x^2+2x^2+11x+11x+18+12)/((2x+3)*(x+2))`

`(3x^2+22x+30)/((2x+3)*(x+2))`

Therefore, the simplified form of our given expression would be
`(3x^2+22x+30)/((2x+3)*(x+2))`.

Answer 2

`(x+9)/(2x+3) + (x+4)/(x+2) \\ = ((x+9)(x+2)+(x+4)(2x+3))/((2x+3)(x+2)) \\ = ( x^(2) +11x+18+2 x^(2) +11x+12)/((2x+3)(x+2)) \\ (3 x^(2) +22x+30)/((2x+3)(x+2))`