If the roots of the equation $\frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1}$ are equal but opposite in sign,then the value of $m$ is

If $b_{1} b_{2} = 2(c_{1} + c_{2})$ and $b_{1}, b_{2}, c_{1}, c_{2}$ are all real numbers, then at least one of the equations $x^{2} + b_{1} x + c_{1} = 0$ and $x^{2} + b_{2} x + c_{2} = 0$ has

The product of all the rational roots of the equation $(x^2-9x+11)^2-(x-4)(x-5)=3$ is equal to:

If the roots of the quadratic equation $ax^2 - bx - c = 0$ are shifted by a constant value,which of the following expressions involving $a, b, c$ remains unchanged?

Let $a, b \in \mathbb{R}, a \neq 0$ be such that the equation $a x^{2}-2 b x+5=0$ has a repeated root $\alpha,$ which is also a root of the equation $x^{2}-2 b x-10=0$. If $\beta$ is the other root of this equation,then $\alpha^{2}+\beta^{2}$ is equal to:

If $\alpha$ and $\beta$ $(\alpha > \beta)$ are the multiple roots of the equation $4x^4 + 4x^3 - 23x^2 - 12x + 36 = 0$,then $2\alpha - \beta = $