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 product of all real roots of the equation ${x^2} - |x| - \,6 = 0$ is
The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is:
Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to :
Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $x ^2-5| x +2|-4=0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^2+ mn + n ^2$ is equal to $..............$.
If $x,\;y,\;z$ are real and distinct, then $u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - zxy$ is always