For what value of $a$, the sum of the squares of the roots of the equation $x^2-(a-2)x-a+1=0$ will have the least value?

If one of the roots of the equation $x^2 + ax + 3 = 0$ is $3$,and one of the roots of the equation $x^2 + ax + b = 0$ is three times the other root,then the value of $b$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\alpha \beta^2 + \alpha^2 \beta + \alpha \beta$?

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - (1 + {n^2})x + \frac{1}{2}(1 + {n^2} + {n^4}) = 0$,then the value of ${\alpha ^2} + {\beta ^2}$ is

In a triangle $PQR$,$\angle R = \frac{\pi}{2}$. If $\tan(\frac{P}{2})$ and $\tan(\frac{Q}{2})$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then:

If in the equation $ax^2 + bx + c = 0$,the sum of roots is equal to the sum of the squares of their reciprocals,then $\frac{c}{a}, \frac{a}{b}, \frac{b}{c}$ are in