Mathematics - Quadratic Equation Question with Solution | TestHub

(3) $$\begin{array}{rl}& (1-p{)}^2+p(1-p)+(1-p)=0\left(\text{ since }(1-p)\text{ is a root of the equation }{x}^2+px+(1-p)=0\right)\\ & \Rightarrow (1-p)(1-p+p+1)=0\\ & \Rightarrow 2(1-p)=0\Rightarrow (1-p)=0\Rightarrow p=1\\ & \text{ sum of root is }\alpha +\beta =-p\text{ and product }\alpha \beta =1-p=0\text{ (where }\beta =1-p=0)\\ & \Rightarrow \alpha +0=-1\Rightarrow \alpha =-1\Rightarrow \text{ Roots are }0,-1\end{array}$$