Find the quadratic equation the sum of whose roots is one and sum of the squares of the roots is 13 
 

Let $\alpha , \beta$ be the roots of a required quadratic equation 
Given 

$$\begin{gathered} \alpha +\beta =1, {\alpha }^2+{\beta }^2=13 \\ {\left(\alpha +\beta \right)}^2-2\alpha \beta =13 \\ 1-2\alpha \beta =13 \\ 2\alpha \beta =-12; \alpha \beta =-6 \\ The required equation is x^2-\left(\alpha +\beta \right)x+\alpha \beta =0 \\ =x^2-x-6=0 \end{gathered}$$