The equation formed by decreasing each root of $ax^2 + bx + c = 0$ by $1$ is $2x^2 + 8x + 2 = 0$. Then:

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

Solve the given two equations and select the correct answer from the given options.
$I. x = \sqrt{1764}$
$II. y^2 = 1764$

If the roots of the equation $x^2 + 2mx + m^2 - 2m + 6 = 0$ are equal,then the value of $m$ is:

If $\alpha_1, \alpha_2$ and $\beta_1, \beta_2$ are the roots of the equations $ax^2 + bx + c = 0$ and $px^2 + qx + r = 0$ respectively,and the system of equations $\alpha_1 y + \alpha_2 z = 0$ and $\beta_1 y + \beta_2 z = 0$ has a non-zero solution,then:

If the ratio of the roots of $ax^2 + 2bx + c = 0$ is the same as the ratio of the roots of $px^2 + 2qx + r = 0$,then