The roots of the equation $x^2 + 2\sqrt{3}x + 3 = 0$ are

For which real value of $a$ do the roots of the quadratic equation $2x^2 - (a^3 + 8a - 1) x + a^2 - 4a = 0$ have opposite signs?

The roots of the cubic equation $3x^3+4x^2-5x-2=0$ are diminished by $h$,and a cubic equation with these diminished roots is formed. If the transformed equation does not contain the $x^2$ term,then the roots of the transformed equation are

If the roots of the equation $(m - n)x^2 + (n - l)x + (l - m) = 0$ are equal,then which of the following is true for $l, m,$ and $n$?

If $x = 2 + 2^{2/3} + 2^{1/3},$ then $x^3 - 6x^2 + 6x = $

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