If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in

Let $a, b$ and $c$ be positive real numbers. If $\frac{x^2-bx}{ax-c} = \frac{m-1}{m+1}$ has two roots which are numerically equal but opposite in sign,then the value of $m$ is

If $\alpha, \beta$ are the roots of the equation $x^2+bx+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$,then $3c+2=$ (in $b$)

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x^{2} + (2 - \lambda)x + (10 - \lambda) = 0$ is minimum,then the magnitude of the difference of the roots of this equation is

If the harmonic mean of the roots of the equation $\sqrt{2} x^2 - bx + (8 - 2\sqrt{5}) = 0$ is $4$,then the value of $b$ is

The difference between two roots of the equation $x^3-13x^2+15x+189=0$ is $2$. Then the roots of the equation are: