Question 125833
{{{4-(a-2)/(a+5)=(a^2-18)/(a+5)}}} Start with the given equation



{{{(a+5)(4-(a-2)/(cross(a+5)))=cross(a+5)((a^2-18)/(cross(a+5)))}}} Multiply both sides by the LCD {{{(a+5)}}}. Doing this will eliminate every fraction.



{{{(a+5)(4)-(a-2)=a^2-18}}} Distribute and multiply. Notice every denominator has been canceled out.



{{{4(a+5)-(a-2)=a^2-18}}} Rearrange the terms



{{{4a+20-a+2=a^2-18}}} Distribute 



{{{3a+22=a^2-18}}} Combine like terms



{{{3a+22-a^2+18=0}}}  Subtract a^2 from both sides.  Add 18 to both sides. 



{{{-a^2+3a+40=0}}} Combine like terms



{{{(-a+8)(a+5)=0}}} Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




Now set each factor equal to zero:

{{{-a+8=0}}} or  {{{a+5=0}}} 


{{{a=8}}} or  {{{a=-5}}}    Now solve for a in each case



So our possible answers are 

 {{{a=8}}} or  {{{a=-5}}} 




However, if you plug in {{{a=-5}}} into the original equation {{{4-(a-2)/(a+5)=(a^2-18)/(a+5)}}}, you'll get a denominator of zero. So {{{a=-5}}} is <b>not</b> a solution since {{{a=-5}}} is not in the domain.




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Answer:


So the only solution is


{{{a=8}}}