Question

Find the roots of the following quadratic equations by using the quadratic formula
$2{\left(\frac{x}{x+1}\right)}^2-5\left(\frac{x}{x+1}\right)+2=0,x\ne -1$

• A. $1,2$
• B. $1,-\frac{1}{2}$
• C. $\frac{1}{2},-2$
• D. $-2,1$

Answer 1

The correct option is D $-2,1$

Given equation is $2{\left(\frac{x}{x+1}\right)}^2-5\left(\frac{x}{x+1}\right)+2=0$

Let $\frac{x}{x+1}=a$

Then $2a^2-5a+2=0$

$\Rightarrow 2a^2-4a-a+2=0$

$\Rightarrow 2a(a-2)-1(a-2)=0$

$\Rightarrow (2a-1)(a-2)=0$

$\therefore a-2=0\Rightarrow a=2$, put the value $a=\frac{x}{x+1}$

If $2a-1=0\Rightarrow a=\frac{1}{2}$, put $a=\frac{x}{x+1}$

$\frac{x}{x+1}=\frac{1}{2}\Rightarrow x+1=2x\Rightarrow x=1$

$\therefore \frac{x}{x+1}=2\Rightarrow x=2x+2\Rightarrow x=-2$