Question 118777


{{{(x+2)(12/x -1)=12}}} Start with the given equation



{{{12x/x-x+24/x-2=12}}} Foil





{{{12-x+24/x-2=12}}} Simplify




{{{x(12-x+24/x-2)=x(12)}}} Multiply both sides by the LCD x to eliminate every fraction.



{{{12x-x^2+24-2x=12x}}} Distribute and multiply



{{{12x-x^2+24-2x-12x=0}}} Subtract 12x from both sides




{{{-x^2-2x+24=0}}} Combine like terms



Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{-x^2-2*x+24=0}}} ( notice {{{a=-1}}}, {{{b=-2}}}, and {{{c=24}}})





{{{x = (--2 +- sqrt( (-2)^2-4*-1*24 ))/(2*-1)}}} Plug in a=-1, b=-2, and c=24




{{{x = (2 +- sqrt( (-2)^2-4*-1*24 ))/(2*-1)}}} Negate -2 to get 2




{{{x = (2 +- sqrt( 4-4*-1*24 ))/(2*-1)}}} Square -2 to get 4  (note: remember when you square -2, you must square the negative as well. This is because {{{(-2)^2=-2*-2=4}}}.)




{{{x = (2 +- sqrt( 4+96 ))/(2*-1)}}} Multiply {{{-4*24*-1}}} to get {{{96}}}




{{{x = (2 +- sqrt( 100 ))/(2*-1)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (2 +- 10)/(2*-1)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (2 +- 10)/-2}}} Multiply 2 and -1 to get -2


So now the expression breaks down into two parts


{{{x = (2 + 10)/-2}}} or {{{x = (2 - 10)/-2}}}


Lets look at the first part:


{{{x=(2 + 10)/-2}}}


{{{x=12/-2}}} Add the terms in the numerator

{{{x=-6}}} Divide


So one answer is

{{{x=-6}}}




Now lets look at the second part:


{{{x=(2 - 10)/-2}}}


{{{x=-8/-2}}} Subtract the terms in the numerator

{{{x=4}}} Divide


So another answer is

{{{x=4}}}


So our solutions are:

{{{x=-6}}} or {{{x=4}}}