$\alpha$ and $\beta$ are the real roots of the equation $12 x^{1/3} - 25 x^{1/6} + 12 = 0$. If $\alpha > \beta$,then $\sqrt[6]{\frac{\alpha}{\beta}} =$

If $P(x)=ax^{2}+bx+c$ and $Q(x)=-ax^{2}+dx+c$, where $ac \neq 0$ ($a, b, c, d$ are all real), then $P(x) \cdot Q(x)=0$ has

Solve the equation $x^{2}+x+\frac{1}{\sqrt{2}}=0$.

Let $p(x) = x^2 + ax + b$ have two distinct real roots,where $a, b$ are real numbers. Define $g(x) = p(x^3)$ for all real numbers $x$. Then,which of the following statements are true?
$I.$ $g$ has exactly two distinct real roots.
$II.$ $g$ can have more than two distinct real roots.
$III.$ There exists a real number $\alpha$ such that $g(x) \geq \alpha$ for all real $x$.

Let $a, b$ and $c$ be the sides of a scalene triangle. If $\lambda$ is a real number such that the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real,then the interval in which $\lambda$ lies is

In a triangle $ABC$,the value of $\angle A$ is given by $5\cos A + 3 = 0$. Find the quadratic equation whose roots are $\sin A$ and $\tan A$.