Question

Roots of the equation
$\frac{2x}{{3x}^2-x+2}-\frac{7x}{{3x}^2+5x+2}=1$ are $\frac{-a-\sqrt{b}}{c}$ and $\frac{-a-\sqrt{b}}{c}$ Find $b-a-c$

Answer 1

Since x=0 is not a root of the equation, then we can divide above and below in each fraction on the left hand side by x and get an equivalent equation, i.e.
$\Rightarrow \frac{2}{3x-x+\frac{2}{x}}-\frac{7}{3x+5x+\frac{2}{x}}=1$
Putting $3x+\frac{2}{x}=y$, then we get
$\Rightarrow \frac{2}{y-1}-\frac{7}{y+5}=1$
$\Rightarrow \frac{17-5y}{(y-1)(y+5)}=1$
$\Rightarrow 17-5y=y^2+4y-5$
$\Rightarrow y^2+9y-22=0$
this equation has two roots $y_1=-11$ and $y_2=2$. Thus the given equation is equivalent to the collection
$3x+\frac{2}{x}=-11$ and $3x+\frac{2}{x}=2$
the second equation of this collection has no roots and the first eqaution has two roots $x_1=\frac{-11+\sqrt{97}}{6}$ and $x_2=\frac{-11-\sqrt{97}}{6}$ which are the solution of the given equation.