If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x - 3 = 0$,then what is the equation whose roots are $\frac{1}{2\alpha - 3}$ and $\frac{1}{2\beta - 3}$?

If $p$ and $q$ are the roots of $x^2 + px + q = 0$,then

Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14 x^2-31 x+3 \lambda=0$ and $\alpha, \gamma$ be the roots of the equation $35 x^2-53 x+4 \lambda=0$. Then $\frac{3 \alpha}{\beta}$ and $\frac{4 \alpha}{\gamma}$ are the roots of the equation:

If the ratio of the roots of the equations $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ are equal,then:

If $\alpha, \beta$ are the roots of $ax^2 + bx + c = 0$ and $\gamma, \delta$ are the roots of $px^2 + qx + r = 0$,and $D_1, D_2$ are their respective discriminants. If $\alpha, \beta, \gamma, \delta$ are in arithmetic progression,then $D_1 : D_2 = \dots$

The value of $a$ such that the sum of the squares of the roots of the equation $x^2 - (a - 2)x - a + 1 = 0$ assumes the least value is: