Question

Solve the given quadratic equation by factorisation method, $\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}$

Answer 1

The given equation $\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}$ can be simplified as follows:

$\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}\Rightarrow \frac{x\left(x\right)+(x+1)(x+1)}{x(x+1)}=\frac{34}{15}\Rightarrow \frac{x^2+\left[x\right(x+1)+1(x+1\left)\right]}{x(x+1)}=\frac{34}{15}\Rightarrow \frac{x^2+(x^2+x+x+1)}{x(x+1)}=\frac{34}{15}\Rightarrow \frac{x^2+x^2+2x+1}{x(x+1)}=\frac{34}{15}\Rightarrow \frac{2x^2+2x+1}{x^2+x}=\frac{34}{15}\Rightarrow 15(2x^2+2x+1)=34(x^2+x)\Rightarrow 30x^2+30x+15=34x^2+34x\Rightarrow 34x^2+34x-30x^2-30x-15=0\Rightarrow 4x^2+4x-15=0$

Now, the quadratic equation $4x^2+4x-15=0$ can be factorised as shown below:

$4x^2+4x-15=0\Rightarrow 4x^2-6x+10x-15=0\Rightarrow 2x(2x-3)+5(2x-3)=0\Rightarrow (2x+5)=0,(2x-3)=0\Rightarrow 2x=-5,2x=3\Rightarrow x=-\frac{5}{2},x=\frac{3}{2}$

Hence, $x=-\frac{5}{2}$ or $x=\frac{3}{2}$.