If $2, 1, 1$ are roots of the equation $x^3-4x^2+5x-2=0$,then find the roots of the equation $\left(x+\frac{1}{3}\right)^3-4\left(x+\frac{1}{3}\right)^2+5\left(x+\frac{1}{3}\right)-2=0$.

If $m, n$ are the roots of the equation ${x^2} - x - 1 = 0$,then the value of $\frac{{\left( {1 + m{{\log }_e}3 + \frac{{{{(m{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)\left( {1 + n{{\log }_e}3 + \frac{{{{(n{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}{{\left( {1 + mn{{\log }_e}3 + \frac{{{{(mn{{\log }_e}3)}^2}}}{{2!}} + ...\infty } \right)}}$ is:

If the roots of the equation $ax^2 + x + b = 0$ are real and distinct,then what will be the nature of the roots of the equation $x^2 - 4\sqrt{ab}x + 1 = 0$?

Solve the equation $27 x^{2}-10 x+1=0$.

Let $f(x) = Ax^2 + Bx$ and $g(x) = Lx^2 + Mx + N$. Given that $f(2) - g(2) = 1$, $f(3) - g(3) = 4$, and $f(4) - g(4) = 9$. Then a root of $f(x) - g(x) = 0$ is

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