Question

its says solve the euation allgebraically. Check for extraneous solutions. The equation is 2 - 3/x+4 = 12/x^2+4x

Answer 1

Ok, so we want to remove the denominators here, and convert this into something we can nicely solve algebraically.

The best way to do this is to multiply by a common denominator, a term which has all of the individual denominators as terms. In this case, that would be (x^2+4x), which can also be written x(x+4).

If we multiply each term by x(x+4) we get:

2x(x+4) - 3x=12
2x^2 + 8x -3x -12 =0
2x^2 + 5x -12 =0

Now factor the result.
(2x-3)(x+4)=0

x COULD be 3/2 and/or -4. We need to check as there could be extraneous roots.

We try substituting x=3/2 and x=-4 in to the original equation. For x=-4, we see how the result gives us something which does not exist, a number over 0.

x=3/2 makes both sides of the equation true, so it is a correct answer.