SOLUTION: Solve: 3 over 2x-3 minus 5 over x = 1

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Question 178228: Solve: 3 over 2x-3 minus 5 over x = 1
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
3 over 2x-3 minus 5 over x = 1
3%2F%282x-3%29+-+5%2Fx+=+1
Start by multiplying both sides by a common denominator x(2x-3):
.
x%282x-3%29%283%2F%282x-3%29+-+5%2Fx%29+=+x%282x-3%29%281%29
.
This results in:
3x+-+5%282x-3%29+=+x%282x-3%29
3x+-+10x+%2B+15+=+2x%5E2+-+3x+
-7x+%2B+15+=+2x%5E2+-+3x+
15+=+2x%5E2+%2B+4x+
0+=+2x%5E2+%2B+4x+-+15
Solve for 'x' using the quadratic equation yields:
x = {1.915, -3.915}
.
Below are the details of the quadratic:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B4x%2B-15+=+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=%284%29%5E2-4%2A2%2A-15=136.

Discriminant d=136 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+136+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+136+%29%29%2F2%5C2+=+1.91547594742265
x%5B2%5D+=+%28-%284%29-sqrt%28+136+%29%29%2F2%5C2+=+-3.91547594742265

Quadratic expression 2x%5E2%2B4x%2B-15 can be factored:
2x%5E2%2B4x%2B-15+=+2%28x-1.91547594742265%29%2A%28x--3.91547594742265%29
Again, the answer is: 1.91547594742265, -3.91547594742265. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B4%2Ax%2B-15+%29