SOLUTION: how do i solve this rational equation? 3x/(x-4) + x/(x+4) = 24/(x^2-16)

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Question 325467: how do i solve this rational equation?
3x/(x-4) + x/(x+4) = 24/(x^2-16)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3x%2F%28x-4%29+%2B+x%2F%28x%2B4%29+=+24%2F%28x%5E2-16%29 Start with the given equation.


3x%2F%28x-4%29+%2B+x%2F%28x%2B4%29+=+24%2F%28%28x-4%29%28x%2B4%29%29 Factor x%5E2-16 to get %28x-4%29%28x%2B4%29


3x%28x%2B4%29+%2B+x%28x-4%29+=+24 Multiply every term by the LCD %28x-4%29%28x%2B4%29 to clear out the fractions.


3x%5E2%2B12x%2B+x%5E2-4x+=+24 Distribute.


3x%5E2%2B12x%2Bx%5E2-4x-24=0 Subtract 24 from both sides.


4x%5E2%2B8x-24=0 Combine like terms.


Notice that the quadratic 4x%5E2%2B8x-24 is in the form of Ax%5E2%2BBx%2BC where A=4, B=8, and C=-24


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%288%29+%2B-+sqrt%28+%288%29%5E2-4%284%29%28-24%29+%29%29%2F%282%284%29%29 Plug in A=4, B=8, and C=-24


x+=+%28-8+%2B-+sqrt%28+64-4%284%29%28-24%29+%29%29%2F%282%284%29%29 Square 8 to get 64.


x+=+%28-8+%2B-+sqrt%28+64--384+%29%29%2F%282%284%29%29 Multiply 4%284%29%28-24%29 to get -384


x+=+%28-8+%2B-+sqrt%28+64%2B384+%29%29%2F%282%284%29%29 Rewrite sqrt%2864--384%29 as sqrt%2864%2B384%29


x+=+%28-8+%2B-+sqrt%28+448+%29%29%2F%282%284%29%29 Add 64 to 384 to get 448


x+=+%28-8+%2B-+sqrt%28+448+%29%29%2F%288%29 Multiply 2 and 4 to get 8.


x+=+%28-8+%2B-+8%2Asqrt%287%29%29%2F%288%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-8%29%2F%288%29+%2B-+%288%2Asqrt%287%29%29%2F%288%29 Break up the fraction.


x+=+-1+%2B-+sqrt%287%29 Reduce.


x+=+-1%2Bsqrt%287%29 or x+=+-1-sqrt%287%29 Break up the expression.


So the solutions are x+=+-1%2Bsqrt%287%29 or x+=+-1-sqrt%287%29