SOLUTION: Find all the solutions to (x + 4)/(x + 5) = (x - 3)/x + (2x - 7)/(x + 5).

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Question 1209258: Find all the solutions to
(x + 4)/(x + 5) = (x - 3)/x + (2x - 7)/(x + 5).
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

The denominators are (x+5) and x
The LCD is x(x+5)

Multiply both sides by the LCD to clear out the fractions.
%28x%2B4%29%2F%28x%2B5%29+=+%28x-3%29%2Fx+%2B+%282x-7%29%2F%28x%2B5%29









x%28x%2B4%29+=+%28x%2B5%29%2A%28x-3%29+%2B+x%282x-7%29

Then let's expand everything out and get everything to one side.
x%28x%2B4%29+=+%28x%2B5%29%2A%28x-3%29+%2B+x%282x-7%29

x%5E2%2B4x+=+%28x%5E2-3x%2B5x-15%29+%2B+%282x%5E2-7x%29

x%5E2%2B4x+=+3x%5E2-5x-15

0+=+3x%5E2-5x-15-x%5E2-4x

0+=+2x%5E2-9x-15

2x%5E2-9x-15+=+0

--------------------------------------------------------------------------

Compare that to ax%5E2%2Bbx%2Bc+=+0
a = 2, b = -9, c = -15

Plug those into the quadratic formula.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-9%29%2B-sqrt%28%28-9%29%5E2-4%282%29%28-15%29%29%29%2F%282%282%29%29

x+=+%289%2B-sqrt%2881+%2B+120%29%29%2F%284%29

x+=+%289%2B-sqrt%28201%29%29%2F%284%29

x+=+%289%2Bsqrt%28201%29%29%2F%284%29+=+5.79436 or x+=+%289-sqrt%28201%29%29%2F%284%29=-1.29436
Those are the two solutions.
The decimal values are approximate.