Question 790553
I get 6 and 10.
You must have made some small mistake.
It is very hard to notice one's own mistakes.
{{{16 / (x+2) = 1 + 2/(x-4)}}}
Multiplying both sides times {{{(x+2)(x-4)}}} I get
{{{16(x+2)(x-4)/ (x+2) = (x+2)(x-4) + 2(x+2)(x-4)/(x-4)}}}
which simplifies to
{{{16(x-4) = (x+2)(x-4) + 2(x+2)}}}
{{{16x-64 = x^2-2x-8 + 2x+4}}}
{{{16x-64 = x^2 -8 +4}}}
{{{16x-64 = x^2-4}}}
{{{0 = x^2-4-16x+64}}}
{{{x^2-16x+60=0}}}
Factoring, I get
{{{(x-6)(x-10)=0}}}
So, either {{{x-6=0}}} --> {{{highlight(x=6)}}},
or {{{x-10=0}}} --> {{{highlight(x=10)}}}.
 
If you do not like factoring, you could use the quadratic formula, or "complete the square" instead.
 
Completing the square:
{{{x^2-16x+60=0}}}
{{{x^2-16x=-60}}}
{{{x^2-16x+64=-60+64}}}
{{{(x-8)^2=4}}}
So either {{{x-8=2}}} --> {{{x=2+8}}} --> {{{highlight(x=10)}}},
or {{{x-8=-2}}} --> {{{x=-2+8}}} --> {{{highlight(x=6)}}}
 
Using the quadratic formula:
The solutions to {{{ax^2+bx+c=0}}} are given by {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}.
In the case of {{{x^2-16x+60=0}}},
{{{a=1}}}, {{{b=-16}}}, and {{{c=60}}}, so
{{{x = (-(-16) +- sqrt((-16)^2-4*1*60 ))/(2*1) =(16 +- sqrt(256-240))/2=(16 +- sqrt(16))/2=(16 +- 4)/2}}}.
The solutions are
{{{x=(16 + 4)/2=20/2=highlight(x=10)}}}
and {{{x=(16 - 4)/2=12/2=highlight(x=6)}}}