SOLUTION: I am trying to solve this equation and I am having a hard time with it... x^-1 + 3/(x+2) = 8 1/2 solve for x

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Question 473699: I am trying to solve this equation and I am having a hard time with it... x^-1 + 3/(x+2) = 8 1/2 solve for x
Found 2 solutions by Alan3354, Edwin McCravy:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
x^-1 + 3/(x+2) = 8 1/2 solve for x
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1/x + 3/(x+2) = 17/2
Multiply by 2x(x+2)
2(x+2) + 6x = 17x(x+2)
17x%5E2+%2B+34x+=+2x%2B+4+%2B+6x
17x%5E2+%2B+26x+-+4+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 17x%5E2%2B26x%2B-4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2826%29%5E2-4%2A17%2A-4=948.

Discriminant d=948 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-26%2B-sqrt%28+948+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2826%29%2Bsqrt%28+948+%29%29%2F2%5C17+=+0.140870842255332
x%5B2%5D+=+%28-%2826%29-sqrt%28+948+%29%29%2F2%5C17+=+-1.67028260696121

Quadratic expression 17x%5E2%2B26x%2B-4 can be factored:
17x%5E2%2B26x%2B-4+=+%28x-0.140870842255332%29%2A%28x--1.67028260696121%29
Again, the answer is: 0.140870842255332, -1.67028260696121. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+17%2Ax%5E2%2B26%2Ax%2B-4+%29

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Is it supposed to be this

x%5E%28-1%29+%2B+3%2F%28x%2B2%29+=+8%261%2F2 ?

or this

%28x%5E%28-1%29+%2B+3%29%2F%28x%2B2%29+=+8%261%2F2 ?

When there aren't enough parentheses to show where numerators
or denominators begin and end, we can't tell what you mean, but
either go by standard rules or whatever rules we think you meant.

If it's this:

x%5E%28-1%29+%2B+3%2F%28x%2B2%29+=+8%261%2F2

1%2Fx+%2B+3%2F%28x%2B2%29+=+17%2F2

Clear of fractions by multiplying through by LCD = 2x(x+2)

2%28x%2B2%29+%2B+6x+=+17x%28x%2B2%29

2x%2B4+%2B+6x+=+17x%5E2+%2B+34x

8x%2B4=17x%5E2%2B34x

0=17x%5E2%2B26x-4

17x%5E2%2B26x-4=0

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

a=17, b=26, c=-4

x+=+%28-26+%2B-+sqrt%28+26%5E2-4%2A17%2A%28-4%29+%29%29%2F%282%2A17%29+

x+=+%28-26+%2B-+sqrt%28676%2B272+%29%29%2F34+

x+=+%28-26+%2B-+sqrt%28948%29%29%2F34+

x+=+%28-26+%2B-+2sqrt%28237%29%29%2F34+

x+=+%282%28-13+%2B-+sqrt%28237%29%29%29%2F34+

x+=+%28-13+%2B-+sqrt%28237%29%29%2F17+
---------------------------------

If it's this:

%28x%5E%28-1%29+%2B+3%29%2F%28x%2B2%29+=+8%261%2F2

%28x%5E%28-1%29+%2B+3%29%2F%28x%2B2%29+=+17%2F2

Multiply through by LCD = 2(x+2)

2%28x%5E%28-1%29+%2B+3%29+=+17%28x%2B2%29

2x%5E%28-1%29+%2B+6+=+17x+%2B+34

2%2Fx+%2B+6+=+17x+%2B+34

Multiply through by LCD = x

2+%2B+6x+=+17x%5E2+%2B+34x

0=17x%5E2+%2B+28x+-+2

x+=+%28-28+%2B-+sqrt%2828%5E2-4%2A17%2A%28-2%29+%29%29%2F%282%2A17%29+

x+=+%28-28+%2B-+sqrt%28784%2B136%29+%29%2F34+

x+=+%28-28+%2B-+sqrt%28920%29+%29%2F34+


x+=+%28-28+%2B-+sqrt%284%2A230%29+%29%2F34+

x+=+%28-28+%2B-+2sqrt%28230%29+%29%2F34+

x+=+%282%28-14+%2B-+sqrt%28230%29%29+%29%2F34+

x+=+%28-14+%2B-+sqrt%28230%29%29+%2F17+

Edwin