Consider the cubic equation $x^3+c x^2+b x+c=0$ where $a, b, c$ are real numbers. Which of the following statements is correct?
If $a^2-2 b < 0$, then the equation has one real and two imaginary roots
If $a^2-2 b \geq 0$, then the equation has all real roots.
If $a^2-2 b > 0$, then the equation has all real and distinct roots.
If $4 a^3-27 b^2 > 0$, then the equation has real and distinct roots.
Let $a$ , $b$ , $c$ are roots of equation $x^3 + 8x + 1 = 0$ ,then the value of
$\frac{{bc}}{{(8b + 1)(8c + 1)}} + \frac{{ac}}{{(8a + 1)(8c + 1)}} + \frac{{ab}}{{(8a + 1)(8b + 1)}}$ is equal to
The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is
Let $t$ be real number such that $t^2=a t+b$ for some positive integers $a$ and $b$. Then, for any choice of positive integers $a$ and $b, t^3$ is never equal to
The sum of all real values of $x$ satisfying the equation ${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}} = 1$ is ;