The roots of $ax^2 + b = 0$ are real and distinct if:

The quadratic equation whose one root is $\frac{1}{2 + \sqrt{5}}$ will be

The number of solutions of the equation $\sqrt{3x^2+x+5} = x-3$ is

The number of all common roots of the equation $x^4-10x^3+37x^2-60x+36=0$ and the transformed equation obtained by increasing any two distinct roots of it by $1$,keeping the other two roots fixed,is

If $\alpha \neq 0$ and $0$ are the roots of the equation $x^2 - 5kx + (6k^2 - 2k) = 0$,then $\alpha = $

The expression $y = ax^2 + bx + c$ always has the same sign as $c$ if