SOLUTION: Need help with solving this equation. 2=(x+3)/(x+2)+5/(x^2-x-6)

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Question 171794: Need help with solving this equation.
2=(x+3)/(x+2)+5/(x^2-x-6)
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
2=%28x%2B3%29%2F%28x%2B2%29%2B5%2F%28x%5E2-x-6%29 Start with the given equation.


2=%28x%2B3%29%2F%28x%2B2%29%2B5%2F%28%28x%2B2%29%28x-3%29%29 Factor x%5E2-x-6 to get %28x%2B2%29%28x-3%29


Take note that the LCD is %28x%2B2%29%28x-3%29



2%28x%2B2%29%28x-3%29=%28%28x%2B3%29%2Fcross%28%28x%2B2%29%29%29%28cross%28%28x%2B2%29%29%28x-3%29%29%2B%285%2F%28cross%28%28x%2B2%29%28x-3%29%29%29%29%28cross%28%28x%2B2%29%28x-3%29%29%29 Multiply EVERY term by the LCD %28x%2B2%29%28x-3%29 to clear out the fractions.


2%28x%2B2%29%28x-3%29=%28x%2B3%29%28x-3%29%2B5 Multiply and simplify


2%28x%5E2-x-6%29=x%5E2-9%2B5 FOIL


2x%5E2-2x-12=x%5E2-9%2B5 Distribute


2x%5E2-2x-12=x%5E2-4 Combine like terms.


2x%5E2-2x-12-x%5E2%2B4=0 Subtract x%5E2 from both sides. Add 4 to both sides.


x%5E2-2x-8=0 Combine like terms.



Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-2, and c=-8


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-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-8%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-2, and c=-8


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-8%29+%29%29%2F%282%281%29%29 Negate -2 to get 2.


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


x+=+%282+%2B-+sqrt%28+4--32+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-8%29 to get -32


x+=+%282+%2B-+sqrt%28+4%2B32+%29%29%2F%282%281%29%29 Rewrite sqrt%284--32%29 as sqrt%284%2B32%29


x+=+%282+%2B-+sqrt%28+36+%29%29%2F%282%281%29%29 Add 4 to 32 to get 36


x+=+%282+%2B-+sqrt%28+36+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%282+%2B-+6%29%2F%282%29 Take the square root of 36 to get 6.


x+=+%282+%2B+6%29%2F%282%29 or x+=+%282+-+6%29%2F%282%29 Break up the expression.


x+=+%288%29%2F%282%29 or x+=++%28-4%29%2F%282%29 Combine like terms.


x+=+4 or x+=+-2 Simplify.


So the possible solutions are x+=+4 or x+=+-2

However, if you plug in x+=+-2 back into the original equation, a division by zero will occur.

So x+=+-2 is NOT a solution.


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Answer:


So the solution is x+=+4