SOLUTION: Help in solving {{{2/5+x/3=1/x}}}

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: Help in solving {{{2/5+x/3=1/x}}}      Log On


   

Question 875278: Help in solving 2%2F5%2Bx%2F3=1%2Fx
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
2%2F5%2Bx%2F3=1%2Fx

We could say we use a common denominator of 5%2A3%2Ax=15x :
2%2F5%2Bx%2F3=1%2Fx
6x%2F15x%2B5x%5E2%2F15x=15%2F15x
%286x%2B5x%5E2%29%2F15x=15%2F15x
6x%2B5x%5E2=15

Or we could say that we multiply times 5%2A3%2Ax=15x :
2%2F5%2Bx%2F3=1%2Fx
15x%282%2F5%2Bx%2F3%29=15x%281%2Fx%29
15x%282%2F5%29%2B15x%28x%2F3%29=15x%2Fx
30x%2F5%2B15x%5E2%2F3=15
6x%2B5x%5E2=15

It's the same thing, we end up with
6x%2B5x%5E2=15 <---> 6x%2B5x%5E2-15=0 <---> 5x%5E2%2B6x-15=0 anyway,
and now we have to solve that quadratic equation.
Factoring does not work, so we have two options to solve that equation:

USING THE QUADRATIC FORMULA:
The quadratic formula says that the solutions (real or imaginary) of
ax%5E2%2Bbx%2Bc=0 are given by
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ .
In the case of 5x%5E2%2B6x-15=0 ,
a=5 , b=6 , and c=-15 , so

That can be simplified as


COMPLETING THE SQUARE:
5x%5E2%2B6x-15=0
5x%5E2%2B6x=15
%285x%5E2%2B6x%29%2F5=15%2F5
x%5E2%2B%286%2F5%29x=3
x%5E2%2B2%283%2F5%29x%2B%283%2F5%29%5E2=3%2B%283%2F5%29%5E2
%28x%2B3%2F5%29%5E2=75%2F25%2B9%2F25
%28x%2B3%2F5%29%5E2=84%2F25
Either
or x%2B3%2F5=-2sqrt%2821%29%2F5
x%2B3%2F5=2sqrt%2821%29%2F5-->x=-3%2F5%2B2sqrt%2821%29%2F5=%28-3%2B2sqrt%2821%29%295
x%2B3%2F5=-2sqrt%2821%29%2F5-->x=-3%2F5-2sqrt%2821%29%2F5=%28-3-2sqrt%2821%29%295
and both solutions can be expressed as x=%28-3+%2B-+2sqrt%2821%29%29%2F5