SOLUTION: SOLVE THE EQUATION 2/x-2+x/x+2=x+6/x^2-4

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Question 170424: SOLVE THE EQUATION
2/x-2+x/x+2=x+6/x^2-4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%282%29%2F%28x-2%29%2B%28x%29%2F%28x%2B2%29=%28x%2B6%29%2F%28x%5E2-4%29 Start with the given equation.


%282%29%2F%28x-2%29%2B%28x%29%2F%28x%2B2%29=%28x%2B6%29%2F%28%28x-2%29%28x%2B2%29%29 Factor x%5E2-4 to get %28x-2%29%28x%2B2%29


Take note that the LCD is %28x-2%29%28x%2B2%29


Multiply every term on both sides by the LCD %28x-2%29%28x%2B2%29. Doing this will eliminate all of the fractions.


2%28x%2B2%29%2Bx%28x-2%29=x%2B6 Multiply and simplify


2x%2B4%2Bx%5E2-2x=x%2B6 Distribute.


2x%2B4%2Bx%5E2-2x-x-6=0 Subtract x from both sides. Subtract 6 from both sides.


x%5E2-x-2=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-1, and c=-2


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-1, and c=-2


x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29 Negate -1 to get 1.


x+=+%281+%2B-+sqrt%28+1-4%281%29%28-2%29+%29%29%2F%282%281%29%29 Square -1 to get 1.


x+=+%281+%2B-+sqrt%28+1--8+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-2%29 to get -8


x+=+%281+%2B-+sqrt%28+1%2B8+%29%29%2F%282%281%29%29 Rewrite sqrt%281--8%29 as sqrt%281%2B8%29


x+=+%281+%2B-+sqrt%28+9+%29%29%2F%282%281%29%29 Add 1 to 8 to get 9


x+=+%281+%2B-+sqrt%28+9+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%281+%2B-+3%29%2F%282%29 Take the square root of 9 to get 3.


x+=+%281+%2B+3%29%2F%282%29 or x+=+%281+-+3%29%2F%282%29 Break up the expression.


x+=+%284%29%2F%282%29 or x+=++%28-2%29%2F%282%29 Combine like terms.


x+=+2 or x+=+-1 Simplify.


So the possible answers are x+=+2 or x+=+-1


However, if you plug in x+=+2, you'll get a division by zero. So x+=+2 is not an answer (since x+=+2 is not even in the domain).



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Answer:


So the solution is x+=+-1