SOLUTION: Solve the rational equation (5/3x+5)-(1/x)=(1/2x+3)

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Question 5530: Solve the rational equation
(5/3x+5)-(1/x)=(1/2x+3)
Answer by Abbey(339) About Me  (Show Source):
You can put this solution on YOUR website!
multiply the entire equation by (3x+5)(x)(2x+3)
This will clear our denominators, because they now cancel:
%28x%29%282x%2B3%295+-1%2A%283x%2B5%29%282x%2B3%29+=+1%2A%283x%2B5%29%28x%29
clear the parentheses:
5x%2A%282x%2B3%29-1%286x%5E2%2B19x%2B15%29=x%283x%2B5%29
%2810x%5E2%2B15x%29-6x%5E2-19x-15=3x%5E2%2B5x
combine like terms:
4x%5E2-4x-15=3x%5E2%2B5x
subtract 5x from both sides:
4x%5E2%2B-9x-15=3x%5E2
Subtract 3x^2 from both sides:
x%5E2-9x-15=0
Then 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%2B-9x%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=%28-9%29%5E2-4%2A1%2A-15=141.

Discriminant d=141 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--9%2B-sqrt%28+141+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-9%29%2Bsqrt%28+141+%29%29%2F2%5C1+=+10.437171043519
x%5B2%5D+=+%28-%28-9%29-sqrt%28+141+%29%29%2F2%5C1+=+-1.43717104351896

Quadratic expression 1x%5E2%2B-9x%2B-15 can be factored:
1x%5E2%2B-9x%2B-15+=+1%28x-10.437171043519%29%2A%28x--1.43717104351896%29
Again, the answer is: 10.437171043519, -1.43717104351896. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-9%2Ax%2B-15+%29