Question

What is the solution set for `8/(x+2)=(x+4)/(x-6)`?

Answer 1

There are no real solutions and two complex solutions `x=1\pm i sqrt(55)`

Explanation:

First, cross multiply to get `8(x-6)=(x+2)(x+4)`. Next, expand to get `8x-48=x^2+6x+8`. Now rearrange to obtain `x^2-2x+56=0`.

The quadratic formula now gives solutions

`x=(2\pm sqrt(4-224))/2=1\pm 1/2 sqrt(-220)`

`=1\pm 1/2 i sqrt(4)sqrt(55)=1\pm isqrt(55)`

These are definitely worth checking in the original equation. I'll check the first and you can check the second.

The left-hand-side of the original equation, upon substitution of `x=1+i sqrt(55)` becomes:

`8/(3+isqrt(55))=(8(3-isqrt(55)))/(9+55)=3/8-i sqrt(55)/8`

Now do the same substitution on the right-hand-side of the original equation:

`(5+isqrt(55))/(-5+isqrt(55))=((5+isqrt(55)) * (-5-isqrt(55)))/(25+55)`

`=(-25-10isqrt(55)+55)/80=3/8-i sqrt(55)/8`

It works! :-)