SOLUTION: I need to solve for x and show work --> (2/x-2)=(4/x-3)-(1/x+1)

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Question 705278: I need to solve for x and show work --> (2/x-2)=(4/x-3)-(1/x+1)
Answer by Stitch(470) About Me  (Show Source):
You can put this solution on YOUR website!
Given:: 2%2F%28X-1%29+=+4%2F%28X-3%29+-+1%2F%28X%2B1%29
I would start by multiplying everything by each of the denominators

Then the denominators will cancel out

Rewrite the equation.
2%2A%28X-3%29%2A%28X%2B1%29+=+4%2A%28X-1%29%2A%28X%2B1%29+-+1%2A%28X-1%29%2A%28X-3%29
Start to multiply through
%282X+-+6%29%2A%28X%2B1%29+=+%284X-4%29%2A%28X%2B1%29+%2B+%28-X%2B1%29%2A%28X-3%29
Multiply through
2X%5E2+%2B+2X+-+6X+-+6+=+4X%5E2+%2B+4X+-+4X+-+4+-+X%5E2+%2B+3X+%2B+X+-+3
Combine like terms
2X%5E2+-+4X+-+6+=+3X%5E2+%2B+4X+-+7
Subtract 2X^2 from both sides
-4X+-+6+=+X%5E2+%2B+4X+-+7
Add 4X to both sides
-6+=+X%5E2+%2B+8X+-+7
Add 6 to both sides
0+=+X%5E2+%2B+8X+-+1
Now use the quadratic equation.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aX%5E2%2BbX%2Bc=0 (in our case 1X%5E2%2B8X%2B-1+=+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=%288%29%5E2-4%2A1%2A-1=68.

Discriminant d=68 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-8%2B-sqrt%28+68+%29%29%2F2%5Ca.

X%5B1%5D+=+%28-%288%29%2Bsqrt%28+68+%29%29%2F2%5C1+=+0.123105625617661
X%5B2%5D+=+%28-%288%29-sqrt%28+68+%29%29%2F2%5C1+=+-8.12310562561766

Quadratic expression 1X%5E2%2B8X%2B-1 can be factored:
1X%5E2%2B8X%2B-1+=+1%28X-0.123105625617661%29%2A%28X--8.12310562561766%29
Again, the answer is: 0.123105625617661, -8.12310562561766. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B8%2Ax%2B-1+%29

So X = 0.1231 & -8.1231