Question

How do you solve `(x^2-x-6)/(x+2)+(x^3+x^2)/x=3`?

Answer 1

`x = -1+-sqrt(7)`

Explanation:

`3 = (x^2-x-6)/(x+2) + (x^3+x^2)/x`

`color(white)(3) = ((x-3)color(red)(cancel(color(black)((x+2)))))/color(red)(cancel(color(black)((x+2)))) + (color(red)(cancel(color(black)(x)))(x^2+x))/color(red)(cancel(color(black)(x)))`

`color(white)(3) = x-3+x^2+x = x^2+2x-3`

Subtract `3` from both ends to get:

`0 = x^2+2x-6`

`color(white)(0) = x^2+2x+1-7`

`color(white)(0) = (x+1)^2-(sqrt(7))^2`

`color(white)(0) = ((x+1)-sqrt(7))((x+1)+sqrt(7))`

`color(white)(0) = (x+1-sqrt(7))(x+1+sqrt(7))`

Hence:

`x = -1+-sqrt(7)`

These are both solutions of the original equation, since neither value causes any denominator to be `0`.