Solve the given two equations and select the correct answer from the given options.
$I.$ $12x + 3y = 14$
$II.$ $4x + 2y = 16$

Let $a, b, c$ be real numbers with $a \ne 0$. If $\alpha$ is a root of $a^2x^2 + bx + c = 0$,$\beta$ is a root of $a^2x^2 - bx - c = 0$,and $0 < \alpha < \beta$,then the equation $a^2x^2 + 2bx + 2c = 0$ has a root $\gamma$ that always satisfies:

Solve the given two equations and select the correct answer from the given options.
$I.$ $91x^2 + 298x + 187 = 0$
$II.$ $247y^2 + 216y - 391 = 0$

The integral values of $a$ for which the quadratic equation $(x - a)(x - 10) + 1 = 0$ has integral roots are

Sum of all distinct integral values of $\alpha$ such that the equation $x^2 - \alpha x + \alpha + 1 = 0$ has integral roots,is equal to-

The condition that one root of the equation $ax^2 + bx + c = 0$ is three times the other is