SOLUTION: x/x+1 + 5/x = 1/x^2+x

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Question 200942: x/x+1 + 5/x = 1/x^2+x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%2F%28x%2B1%29%2B5%2Fx=1%2F%28x%5E2%2Bx%29 Start with the given equation.


x%2F%28x%2B1%29%2B5%2Fx=1%2F%28x%28x%2B1%29%29 Factor the last denominator


Multiply EVERY term by the LCD x%28x%2B1%29 to clear out the fractions.


x%2Ax%2B5%28x%2B1%29=1 Simplify


x%2Ax%2B5%2Ax%2B5%2A1=1 Distribute


x%5E2%2B5x%2B5=1 Multiply


x%5E2%2B5x%2B5-1=0 Subtract 1 from both sides.


x%5E2%2B5x%2B4=0 Combine like terms.


Notice that the quadratic x%5E2%2B5x%2B4 is in the form of Ax%5E2%2BBx%2BC where A=1, B=5, and C=4


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-%285%29+%2B-+sqrt%28+%285%29%5E2-4%281%29%284%29+%29%29%2F%282%281%29%29 Plug in A=1, B=5, and C=4


x+=+%28-5+%2B-+sqrt%28+25-4%281%29%284%29+%29%29%2F%282%281%29%29 Square 5 to get 25.


x+=+%28-5+%2B-+sqrt%28+25-16+%29%29%2F%282%281%29%29 Multiply 4%281%29%284%29 to get 16


x+=+%28-5+%2B-+sqrt%28+9+%29%29%2F%282%281%29%29 Subtract 16 from 25 to get 9


x+=+%28-5+%2B-+sqrt%28+9+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-5+%2B-+3%29%2F%282%29 Take the square root of 9 to get 3.


x+=+%28-5+%2B+3%29%2F%282%29 or x+=+%28-5+-+3%29%2F%282%29 Break up the expression.


x+=+%28-2%29%2F%282%29 or x+=++%28-8%29%2F%282%29 Combine like terms.


x+=+-1 or x+=+-4 Simplify.


So the possible solutions are x+=+-1 or x+=+-4


However, recall that x%3C%3E-1, this means that the only solution is x+=+-4

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Answer:

So the solution is x+=+-4