The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

If ${x^2} + px + 1$ is a factor of the expression $a{x^3} + bx + c$, then

In a cubic equation coefficient of $x^2$ is zero and remaining coefficient are real has one root $\alpha = 3 + 4\, i$ and remaining roots are $\beta$ and $\gamma$ then $\alpha \beta \gamma$ is :-

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?

$I$. It has at least one negative root.

$II$. It has at least one positive root.

$III$. Both its roots are real.

lf $2 + 3i$ is one of the roots of the equation $2x^3 -9x^2 + kx- 13 = 0,$ $k \in R,$ then the real root of this equation

The number of real solutions of the equation $|x{|^2}$-$3|x| + 2 = 0$ are