If the roots of the equation $bx^2 + cx + a = 0$ are imaginary,then for all real values of $x,$ the expression $3b^2x^2 + 6bcx + 2c^2$ is :

The real roots of the equation $x^2 + 5|x| + 4 = 0$ are

Statement-$I$: If $a + b + c = 0$ and $a, b, c$ are rational,then the roots of the equation $(b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0$ are rational.
Statement-$II$: The discriminant of $(b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0$ is a perfect square.

If $a, b, c$ are in Arithmetic Progression $(AP)$,then the roots of the equation $ax^2 - 2bx + c = 0$ are

The roots of the equation $(x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0$ for $a \in R$ are always:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4x^2-9x-36=0$ and $\alpha < \beta < \gamma$,then find the value of $\alpha+2\beta+3\gamma$.