Question

Solve `1/x + 1/(x+4) = 1/3` ?

These are all fractions and I know I have to create a common denominator.

Answer 1

`x = 1+-sqrt(13)`

Explanation:

Given:

`1/x+1/(x+4)=1/3`

Multiply both sides of the equation by `3x(x+4)` to get:

`3(x+4)+3x=x(x+4)`

which multiplies out to:

`3x+12+3x=x^2+4x`

That is:

`6x+12 = x^2+4x`

Subtract `6x+12` from both sides to get:

`0 = x^2-2x-12`

`color(white)(0) = x^2-2x+1-13`

`color(white)(0) = (x-1)^2-(sqrt(13))^2`

`color(white)(0) = ((x-1)-sqrt(13))((x-1)+sqrt(13))`

`color(white)(0) = (x-1-sqrt(13))(x-1+sqrt(13))`

So:

`x = 1+-sqrt(13)`

These are both solutions of the original equation since they do not cause any of the denominators to becomes zero.