# Thread: problems on solving rational equation

1. ## problems on solving rational equation

so i've been given a question :
5x / x-4 = 3x/2x + 7

and i had to solve for x, or the root u can say. I've gotten to the step where i cross multiply and bring everything to one side. And i got
7x^2 + 42 x, i'm not sure from here on how to solve for x or the roots.

i also have another rational problem like this :
x / x -2 = x^2-4x + 1/ x-3
and i have gotten to the step of bring everything to one side :
x^3 - 7x^2 + 12x -2 i'm not sure is this correct. but my teacher told me to solve with polynomial divison but i cannot find anything sets that polynomial to 0. so please help :S

2. $\displaystyle \frac{5x}{x-4} = \frac{3x}{2x + 7}$
$\displaystyle 5x \dot (2x+7) = 3x \cdot (x-4)$
$\displaystyle 10x^2 + 35x = 3x^2 - 12x$
$\displaystyle 7x^2 + 47x = 0$
Divide everything by x
$\displaystyle 7x + 47 = 0$
$\displaystyle x = -\frac{47}{7}$

3. $\displaystyle \frac{x}{x -2} = \frac{x^2-4x + 1}{x-3}$
$\displaystyle (x) \cdot (x-3) = (x^2-4x+1) \cdot (x-2)$
$\displaystyle x^2 - 3x = x^3 - 4x^2+x-2x^2+8x-2$
$\displaystyle 0 = x^3 - 7x^2+12x-2$
This has no rational roots.
Let $\displaystyle x = z - \frac{a}{3}$
Equation take form $\displaystyle 0 = z^3 + pz + q$
That you can solve with a long messy method Cubic function - Wikipedia, the free encyclopedia
Sorry there is not always a simple way.