SOLUTION: I'm having some trouble getting the other value of "x" on this problem, Can you help me? Find all real solutions of the equation. (If there are extra answer boxes, enter NONE in

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I'm having some trouble getting the other value of "x" on this problem, Can you help me? Find all real solutions of the equation. (If there are extra answer boxes, enter NONE in      Log On


   

Question 196572: I'm having some trouble getting the other value of "x" on this problem, Can you help me?
Find all real solutions of the equation. (If there are extra answer boxes, enter NONE in the last boxes.
x+6/x-2 = 3/x+2 + 32/x^2-4
thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2B6%29%2F%28x-2%29+=+3%2F%28x%2B2%29%2B32%2F%28x%5E2-4%29 Start with the given equation.


%28x%2B6%29%2F%28x-2%29+=+3%2F%28x%2B2%29%2B32%2F%28%28x-2%29%28x%2B2%29%29 Factor the last denominator. Take note that x%3C%3E-2 or x%3C%3E2 (these values would cause a division by zero)


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



%28x%2B2%29%28x%2B6%29=%28x-2%29%283%29%2B32 Cancel out and simplify


%28x%2B2%29%28x%2B6%29=3%28x-2%29%2B32 Rearrange the terms.


x%5E2%2B8x%2B12=3%28x-2%29%2B32 FOIL


x%5E2%2B8x%2B12=3x-6%2B32 Distribute.


x%5E2%2B8x%2B12-3x%2B6-32=0 Get all terms to the left side.


x%5E2%2B5x-14=0 Combine like terms.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc=0 where a=1, b=5, and c=-14


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-%285%29+%2B-+sqrt%28+%285%29%5E2-4%281%29%28-14%29+%29%29%2F%282%281%29%29 Plug in a=1, b=5, and c=-14


x+=+%28-5+%2B-+sqrt%28+25-4%281%29%28-14%29+%29%29%2F%282%281%29%29 Square 5 to get 25.


x+=+%28-5+%2B-+sqrt%28+25--56+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-14%29 to get -56


x+=+%28-5+%2B-+sqrt%28+25%2B56+%29%29%2F%282%281%29%29 Rewrite sqrt%2825--56%29 as sqrt%2825%2B56%29


x+=+%28-5+%2B-+sqrt%28+81+%29%29%2F%282%281%29%29 Add 25 to 56 to get 81


x+=+%28-5+%2B-+sqrt%28+81+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-5+%2B-+9%29%2F%282%29 Take the square root of 81 to get 9.


x+=+%28-5+%2B+9%29%2F%282%29 or x+=+%28-5+-+9%29%2F%282%29 Break up the expression.


x+=+%284%29%2F%282%29 or x+=++%28-14%29%2F%282%29 Combine like terms.


x+=+2 or x+=+-7 Simplify.


So the possible answers are x+=+2 or x+=+-7


However, recall that we stated earlier that x%3C%3E2 (this would cause a division by zero). So x+=+2 is NOT a solution.


This means that the only solution is x+=+-7