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

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 $b^2-4ac=0$,

$(\lambda-5)(\lambda+1)=0$

$Either, \lambda-5=0,\\ \lambda=5$

$or, \lambda+1=0\\ \Rightarrow \lambda=-1$

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

Hence, the given equation has coincident roots for $\lambda =5$ or $-1$