SOLUTION: Find all solutions, real or complex, to the following equation {{{ 12/(x+1) - 10/(x+3) -2 = 0 }}}

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Question 470056: Find all solutions, real or complex, to the following equation
+12%2F%28x%2B1%29+-+10%2F%28x%2B3%29+-2+=+0+
Found 2 solutions by esmy1234, Alan3354:
Answer by esmy1234(2) About Me  (Show Source):
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
+12%2F%28x%2B1%29+-+10%2F%28x%2B3%29+-2+=+0+
+12%2F%28x%2B1%29+-+10%2F%28x%2B3%29+=+2+
Multiply by (x+1)*(x+3)
12%28x%2B3%29+-+10%28x%2B1%29+=+2%28x%5E2%2B4x%2B4%29+=++2x%5E2+%2B+8x+%2B+8
12x%2B36+-+10x-10+=++2x%5E2+%2B+8x+%2B+8
2x%2B26+=++2x%5E2+%2B+8x+%2B+8
2x%5E2+%2B+6x+-+18+=+0
x%5E2+%2B+3x+-+9+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B3x%2B-9+=+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=%283%29%5E2-4%2A1%2A-9=45.

Discriminant d=45 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-3%2B-sqrt%28+45+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+45+%29%29%2F2%5C1+=+1.85410196624968
x%5B2%5D+=+%28-%283%29-sqrt%28+45+%29%29%2F2%5C1+=+-4.85410196624968

Quadratic expression 1x%5E2%2B3x%2B-9 can be factored:
1x%5E2%2B3x%2B-9+=+%28x-1.85410196624968%29%2A%28x--4.85410196624968%29
Again, the answer is: 1.85410196624968, -4.85410196624968. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-9+%29