SOLUTION: {{{x/(2x+2)+1/4=-1/(x+2)}}}

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Question 270261: x%2F%282x%2B2%29%2B1%2F4=-1%2F%28x%2B2%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%2F%282x%2B2%29%2B1%2F4=-1%2F%28x%2B2%29 Start with the given equation.


x%2F%282%28x%2B1%29%29%2B1%2F4=-1%2F%28x%2B2%29 Factor the first denominator.


Take note that the LCD is 4%28x%2B1%29%28x%2B2%29


Multiply EVERY term by the LCD 4%28x%2B1%29%28x%2B2%29 to eliminate the fractions.


2%28x%2B2%29x%2B%28x%2B1%29%28x%2B2%29=4%28x%2B1%29%28-1%29 Simplify.


2x%28x%2B2%29%2B%28x%2B1%29%28x%2B2%29=4%28-1%29%28x%2B1%29 Rearrange the terms.


2x%28x%2B2%29%2B%28x%2B1%29%28x%2B2%29=-4%28x%2B1%29 Multiply.


2x%5E2%2B4x%2B%28x%2B1%29%28x%2B2%29=-4x-4 Distribute.


2x%5E2%2B4x%2Bx%5E2%2B3x%2B2=-4x-4 FOIL.


2x%5E2%2B4x%2Bx%5E2%2B3x%2B2%2B4x%2B4=0 Get every term to one side.


3x%5E2%2B11x%2B6=0 Combine like terms.


Now let's use the quadratic formula to solve 3x%5E2%2B11x%2B6=0


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 3%2Ax%5E2%2B11%2Ax%2B6=0 ( notice a=3, b=11, and c=6)





x+=+%28-11+%2B-+sqrt%28+%2811%29%5E2-4%2A3%2A6+%29%29%2F%282%2A3%29 Plug in a=3, b=11, and c=6




x+=+%28-11+%2B-+sqrt%28+121-4%2A3%2A6+%29%29%2F%282%2A3%29 Square 11 to get 121




x+=+%28-11+%2B-+sqrt%28+121%2B-72+%29%29%2F%282%2A3%29 Multiply -4%2A6%2A3 to get -72




x+=+%28-11+%2B-+sqrt%28+49+%29%29%2F%282%2A3%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-11+%2B-+7%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-11+%2B-+7%29%2F6 Multiply 2 and 3 to get 6


So now the expression breaks down into two parts


x+=+%28-11+%2B+7%29%2F6 or x+=+%28-11+-+7%29%2F6


Lets look at the first part:


x=%28-11+%2B+7%29%2F6


x=-4%2F6 Add the terms in the numerator

x=-2%2F3 Divide


So one answer is

x=-2%2F3




Now lets look at the second part:


x=%28-11+-+7%29%2F6


x=-18%2F6 Subtract the terms in the numerator

x=-3 Divide


So another answer is

x=-3


So our solutions are:

x=-2%2F3 or x=-3





So the solutions to the equation x%2F%282x%2B2%29%2B1%2F4=-1%2F%28x%2B2%29 are

x=-2%2F3 or x=-3