# How do you solve #3/(5-x)+2/(4-x) = 8/(x+2)# ?

##### 2 Answers

#### Answer:

#### Explanation:

The quickest way to find at least one solution is to "guess":

Note that

Then we find:

#3/(5-color(blue)(2)) + 2/(4-color(blue)(2)) = 3/3+2/2 = 1+1 = 2 = 8/4 = 8/(color(blue)(2)+2)#

Are there any other solutions?

Given:

#3/(5-x)+2/(4-x) = 8/(x+2)#

Subtract

#0 = 8/(x+2)-3/(5-x)-2/(4-x)#

#color(white)(0) = 8/(x+2)+3/(x-5)+2/(x-4)#

Multiply through by

#0 = 8(x-5)(x-4)+3(x+2)(x-4)+2(x+2)(x-5)#

#color(white)(0) = 8(x^2-9x+20)+3(x^2-2x-8)+2(x^2-3x-10)#

#color(white)(0) = (8x^2-72x+160)+(3x^2-6x-24)+(2x^2-6x-20)#

#color(white)(0) = 13x^2-84x+116#

We can solve this quadratic by any method we like, but we already know that

#13x^2-84x+116 = (x-2)(13x-58)#

So the other solution is

#### Answer:

#### Explanation:

Hence