Question

Let us consider a quadratic equation $x^2+\lambda x+\lambda +1.25=0$, where $\lambda$ is a constant.

The value of $\lambda$ such that the above quadratic equation has two coincident roots

• A. $\lambda =5$ or $\lambda =-1$
• B. $\lambda =1$ or $\lambda =5$
• C. $\lambda =-5$ or $\lambda =1$
• D. None of these

Answer 1

Answer: (A)

$\lambda =5$ or $\lambda =-1$

The given equation is
$x^2+\lambda x+\lambda +1.25=0$
$a=1,b=\lambda ,c=\lambda +1.25$
$b^2-4ac={\lambda }^2-4\times 1.(\lambda +1.25)$
$={\lambda }^2-4\lambda -5=(\lambda -5)(\lambda +1)$

The equation has two coincident roots if $b^2-4ac=0$,
$(\lambda -5)(\lambda +1)=0$
$\Rightarrow$ Either $\lambda -5=0,\lambda =5$
$\Rightarrow \lambda +1=0\Rightarrow \lambda =-1$
$\therefore \lambda =5$ or $-1$
Hence, the given equation has coincident roots for $\lambda =5$ or $-1$