SOLUTION: {{{x^2/(x^2-4) = x/(x+2) - 2x/(2-x)}}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: {{{x^2/(x^2-4) = x/(x+2) - 2x/(2-x)}}}      Log On


   

Question 5190: x%5E2%2F%28x%5E2-4%29+=+x%2F%28x%2B2%29+-+2x%2F%282-x%29
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2F%28x%5E2-4%29+=+x%2F%28x%2B2%29+-+2x%2F%282-x%29
add polynomial fractions
x%5E2%2F%28x%5E2-4%29+=+%28x%2A%282-x%29-2x%2A%28x%2B2%29%29%2F%28x%2B2%29%2F%28x-2%29

x%5E2%2F%28x%5E2-4%29+=+%28x%2A%282-x%29-2x%2A%28x%2B2%29%29%2F%28x%5E2-4%29
Remove x^2-4 from both sides, being mindful that it cannot be zero, or that x cannot be +2 or - 2.
x%5E2+=+x%282-x%29+-+2x%28x%2B2%29
x%5E2+=+2x-x%5E2+-+2x%5E2+-+4x
4x%5E2+%2B2x+=+0
4x%5E2%2Bx+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B1x%2B0+=+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=%281%29%5E2-4%2A4%2A0=1.

Discriminant d=1 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+1+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+1+%29%29%2F2%5C4+=+0
x%5B2%5D+=+%28-%281%29-sqrt%28+1+%29%29%2F2%5C4+=+-0.25

Quadratic expression 4x%5E2%2B1x%2B0 can be factored:
4x%5E2%2B1x%2B0+=+%28x-0%29%2A%28x--0.25%29
Again, the answer is: 0, -0.25. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B1%2Ax%2B0+%29


That's your answer. PLEASE double check my work.