All the roots of the equation $x^5+15x^4+94x^3+305x^2+507x+353=0$ are increased by some real number $k$ in order to eliminate the $4^{th}$ degree term from the equation. Now,the coefficient of $x$ in the transformed equation is

The sum of all the roots of the equation $|x^2-8x+15|-2x+7=0$ is:

Let $x, y, z$ be positive real numbers. Which of the following conditions imply $x=y=z$?
$I.$ $x^3+y^3+z^3=3xyz$
$II.$ $x^3+y^2z+yz^2=3xyz$
$III.$ $x^3+y^2z+z^2x=3xyz$
$IV.$ $(x+y+z)^3=27xyz$

Let $p, q$ be real numbers. If $\alpha$ is a root of $x^{2}+3 p^{2} x+5 q^{2}=0$, $\beta$ is a root of $x^{2}+9 p^{2} x+15 q^{2}=0$ and $0 < \alpha < \beta$, then the equation $x^{2}+6 p^{2} x+10 q^{2}=0$ has a root $\gamma$ that always satisfies:

Suppose $p, q, r$ are real numbers such that $q=p(4-p)$,$r=q(4-q)$,and $p=r(4-r)$. The maximum possible value of $p+q+r$ is

Let $a, b \in \mathbb{R}$ be such that the equation $ax^{2}-2bx+15=0$ has a repeated root $\alpha$. If $\alpha$ and $\beta$ are the roots of the equation $x^{2}-2bx+21=0$,then $\alpha^{2}+\beta^{2}$ is equal to