Question

Solve for $x:\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}$

Answer 1

We have,

$\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}$

$\Rightarrow \frac{x^2+{(x+1)}^2}{x(x+1)}=\frac{34}{15}$

$\Rightarrow \frac{x^2+x^2+1+2x}{x(x+1)}=\frac{34}{15}$

$\Rightarrow \frac{2x^2+2x+1}{x^2+x}=\frac{34}{15}$

$\Rightarrow 30x^2+30x+15=34x^2+34x$

$\Rightarrow 30x^2-34x^2+30x-34x+15=0$

$\Rightarrow -4x^2-4x+15=0$

$\Rightarrow 4x^2+4x-15=0$

$\Rightarrow 4x^2+(10-6)x-15=0$

$\Rightarrow 4x^2+10x-6x-15=0$

$\Rightarrow 2x(2x+5)-3(2x+5)=0$

$\Rightarrow (2x+5)(2x-3)=0$

$\therefore x=\frac{-5}{2},\frac{3}{2}$