If $a < 0$ then the inequality $a{x^2} - 2x + 4 > 0$ has the solution represented by
$\frac{{1 + \sqrt {1 - 4a} }}{a} > x > \frac{{1 - \sqrt {1 - 4a} }}{a}$
$x < \frac{{1 - \sqrt {1 - 4a} }}{a}$
$x < 2$
$2 > x > \frac{{1 + \sqrt {1 - 4a} }}{a}$
The real roots of the equation ${x^2} + 5|x| + \,\,4 = 0$ are
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
If $\alpha $, $\beta$, $\gamma$ are roots of ${x^3} - 2{x^2} + 3x - 2 = 0$ , then the value of$\left( {\frac{{\alpha \beta }}{{\alpha + \beta }} + \frac{{\alpha \gamma }}{{\alpha + \gamma }} + \frac{{\beta \gamma }}{{\beta + \gamma }}} \right)$ is
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
Suppose that $x$ and $y$ are positive number with $xy = \frac{1}{9};\,x\left( {y + 1} \right) = \frac{7}{9};\,y\left( {x + 1} \right) = \frac{5}{{18}}$ . The value of $(x + 1) (y + 1)$ equals