Problem Solving

indu.nair wrote:Find x if:

3/x+3 + 4/x+4 = 2/x+2 + 5/x+5


There should be some fast way to do this. Anyone??[/list]

I am assuming the question is 3/(x + 3) + 4/(x + 4) = 2/(x + 2) + 5/(x + 5)

In case the answer choices were given to us, then we would have plugged the value to find which value of x satisfies the given equation, but since the answer choices are not known so we can solve this by the method explained below.

Algebraic Method:

{3(x + 4) + 4(x + 3)}/(x + 3)(x + 4) = {2(x + 5) + 5(x + 2)}/(x + 2)(x + 5)
{3x + 12 + 4x + 12}/(x² + 7x + 12) = {2x + 10 + 5x + 10}/(x² + 7x + 10)
{7x + 24}/(x² + 7x + 12) = {7x + 20}/(x² + 7x + 10)
(7x + 24)(x² + 7x + 10) = (7x + 20)(x² + 7x + 12)
7x^3 + 24x² + 49x² + 168x + 70x + 240 = 7x^3 + 20x² + 49x² + 140x + 84x + 240
4x² + 14x = 0
2x² + 7x = 0
x(2x + 7) = 0
x = 0, -7/2