The equation $\sqrt {3 {x^2} + x + 5} = x - 3$ , where $x$ is real, has

Let $p(x)=x^2-5 x+a$ and $q(x)=x^2-3 x+b$, where $a$ and $b$ are positive integers. Suppose HCF $(p(x), q(x))=x-1$ and $k(x)=1 cm (p(x), q(x))$ If the coefficient of the highest degree term of $k(x)$ is 1 , then sum of the roots of $(x-1)+k(x)$ is

The two roots of an equation ${x^3} - 9{x^2} + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots will be

The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is:

If the sum of two of the roots of ${x^3} + p{x^2} + qx + r = 0$ is zero, then $pq =$

Let $a, b, c$ be non-zero real roots of the equation $x^3+a x^2+b x+c=0$. Then,