For what condition will the expression $a^2x^2 + bx + 1$ be positive for all $x \in R$?

If $x$ is real,then the value of $\frac{x^2-3x+4}{x^2+3x+4}$ lies in the interval

Let $x$ and $y$ be two positive real numbers such that $x+y=1$. Then,the minimum value of $\frac{1}{x}+\frac{1}{y}$ is

If the expression $7+6x-3x^2$ attains its extreme value $\beta$ at $x=\alpha$,then the sum of the squares of the roots of the equation $x^2+\alpha x-\beta=0$ is

If the minimum value of $f(x)=2x^2+\alpha x+8$ is the same as the maximum value of $g(x)=-3x^2-4x+\alpha^2$,then $\alpha^2=$

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: