The expression $ax^{2} + bx + c$ (where $a, b,$ and $c$ are real numbers) has the same sign as that of $a$ for all $x \in \mathbb{R}$ if:

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.

How many roots does the equation $x - \frac{2}{x - 1} = 1 - \frac{2}{x - 1}$ have?

The roots of the equation $x^4+x^3-4x^2+x+1=0$ are diminished by $h$ so that the transformed equation does not contain the $x^2$ term. If the values of such $h$ are $\alpha$ and $\beta$,then $12(\alpha-\beta)^2=$

If $a, b$ and $c$ are positive numbers in a $GP$, then the roots of the quadratic equation $(\log_{e} a) x^{2} - (2 \log_{e} b) x + (\log_{e} c) = 0$ are

If the difference between the roots of the equations $x^2 - bx + c = 0$ and $x^2 - cx + b = 0$ is the same,then $b + c = \dots$