SOLUTION: Can you please show me how to solve the following equation. We are not solving for x, but for all solutions. x/x-1+8/x=10/4

Algebra ->  Equations -> SOLUTION: Can you please show me how to solve the following equation. We are not solving for x, but for all solutions. x/x-1+8/x=10/4      Log On


   

Question 681678: Can you please show me how to solve the following equation. We are not solving for x, but for all solutions.
x/x-1+8/x=10/4
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Can you please show me how to solve the following equation. We are not solving for x, but for all solutions.
-------
x is the only variable, the only thing to solve for.
----------
x/(x-1) + 8/x = 10/4 = 5/2
Mutliply by 2x(x-1)
2x*x + 8*2*(x-1) = 5x*(x-1)
2x%5E2+%2B+16x+-+16+=+5x%5E2+-+5x
3x%5E2+-+21x+%2B+16+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-21x%2B16+=+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=%28-21%29%5E2-4%2A3%2A16=249.

Discriminant d=249 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--21%2B-sqrt%28+249+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-21%29%2Bsqrt%28+249+%29%29%2F2%5C3+=+6.12995563967658
x%5B2%5D+=+%28-%28-21%29-sqrt%28+249+%29%29%2F2%5C3+=+0.870044360323417

Quadratic expression 3x%5E2%2B-21x%2B16 can be factored:
3x%5E2%2B-21x%2B16+=+%28x-6.12995563967658%29%2A%28x-0.870044360323417%29
Again, the answer is: 6.12995563967658, 0.870044360323417. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-21%2Ax%2B16+%29