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 distinct roots

• A. $\lambda <5$
• B. $\lambda >-1$
• C. $\lambda >5$
• D. $\lambda <-1$

Answer 1

Answer: (C), (D)

The correct options are
C $\lambda <-1$
D $\lambda >5$

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