SOLUTION: find all real or imaginary solutions to each equation. x-1/x+2 = 2x-3/x+4

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Question 96449: find all real or imaginary solutions to each equation.
x-1/x+2 = 2x-3/x+4

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-1%29%2F%28x%2B2%29+=+%282x-3%29%2F%28x%2B4%29 Start with the given equation


Multiply both sides by the LCD

Notice the LCD eliminates the denominators


%28x%2B4%29%28x-1%29+=+%28x%2B2%29%282x-3%29 Simplify


x%5E2%2B3x-4+=+2x%5E2%2Bx-6 Foil both sides


0=+x%5E2-2x-2 Get all terms to one side and combine like terms



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 x%5E2-2%2Ax-2=0 ( notice a=1, b=-2, and c=-2)

x+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A-2+%29%29%2F%282%2A1%29 Plug in a=1, b=-2, and c=-2



x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A-2+%29%29%2F%282%2A1%29 Negate -2 to get 2



x+=+%282+%2B-+sqrt%28+4-4%2A1%2A-2+%29%29%2F%282%2A1%29 Square -2 to get 4 (note: remember when you square -2, you must square the negative as well. This is because %28-2%29%5E2=-2%2A-2=4.)



x+=+%282+%2B-+sqrt%28+4%2B8+%29%29%2F%282%2A1%29 Multiply -4%2A-2%2A1 to get 8



x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%282+%2B-+2%2Asqrt%283%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%282+%2B-+2%2Asqrt%283%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%282+%2B+2%2Asqrt%283%29%29%2F2 or x+=+%282+-+2%2Asqrt%283%29%29%2F2


Now break up the fraction


x=%2B2%2F2%2B2%2Asqrt%283%29%2F2 or x=%2B2%2F2-2%2Asqrt%283%29%2F2


Simplify


x=1%2Bsqrt%283%29 or x=1-sqrt%283%29


So these expressions approximate to

x=2.73205080756888 or x=-0.732050807568877


So our solutions are:
x=2.73205080756888 or x=-0.732050807568877

Notice when we graph x%5E2-2%2Ax-2, we get:



when we use the root finder feature on a calculator, we find that x=2.73205080756888 and x=-0.732050807568877.So this verifies our answer