Let $a, b, c$ be the length of three sides of a triangle satisfying the condition $\left(a^2+b^2\right) x^2-2 b(a+c)$. $x+\left(b^2+c^2\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12\left(\alpha^2+\beta^2\right)$ is equal to............................

If $x$ be real, then the minimum value of ${x^2} - 8x + 17$ is

Let $f(x)=x^4+a x^3+b x^2+c$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $i \sqrt{3}$ is a root of the equation $4 x^3+3 a x^2+2 b x=0$, where $i=\sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x)=0$, then $\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2$ is equal to. . . . . .

In the equation ${x^3} + 3Hx + G = 0$, if $G$ and $H$ are real and ${G^2} + 4{H^3} > 0,$ then the roots are

If $a, b, c, d$ are four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$, then the minimum value of $\frac{a}{b}+\frac{c}{d}$ is

Sum of the solutions of the equation $\left[ {{x^2}} \right] - 2x + 1 = 0$ is (where $[.]$ denotes greatest integer function)