SOLUTION: 2/x+2/x-1=1

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Question 99578: 2/x+2/x-1=1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Fx%2B2%2F%28x-1%29=1 Start with the given equation


x%28x-1%29%282%2Fx%2B2%2F%28x-1%29%29=x%28x-1%29%281%29 Multiply both sides by the LCD x%28x%2B1%29


2%28x-1%29%2B2x=x%28x-1%29 Distribute and multiply


2x-2%2B2x=x%5E2-x Distribute again


0=x%5E2-5x%2B2 Get all terms to one side


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


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve 0=x%5E2-5%2Ax%2B2 ( notice a=1, b=-5, and c=2)

x+=+%28--5+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 Plug in a=1, b=-5, and c=2



x+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 Negate -5 to get 5



x+=+%285+%2B-+sqrt%28+25-4%2A1%2A2+%29%29%2F%282%2A1%29 Square -5 to get 25 (note: remember when you square -5, you must square the negative as well. This is because %28-5%29%5E2=-5%2A-5=25.)



x+=+%285+%2B-+sqrt%28+25%2B-8+%29%29%2F%282%2A1%29 Multiply -4%2A2%2A1 to get -8



x+=+%285+%2B-+sqrt%28+17+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%285+%2B-+sqrt%2817%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%285+%2B-+sqrt%2817%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%285+%2B+sqrt%2817%29%29%2F2 or x+=+%285+-+sqrt%2817%29%29%2F2


Now break up the fraction


x=%2B5%2F2%2Bsqrt%2817%29%2F2 or x=%2B5%2F2-sqrt%2817%29%2F2


Simplify


x=5+%2F+2%2Bsqrt%2817%29%2F2 or x=5+%2F+2-sqrt%2817%29%2F2


So these expressions approximate to

x=4.56155281280883 or x=0.43844718719117


So our solutions are:
x=4.56155281280883 or x=0.43844718719117

Notice when we graph x%5E2-5%2Ax%2B2, we get:



when we use the root finder feature on a calculator, we find that x=4.56155281280883 and x=0.43844718719117.So this verifies our answer