Quadratic Equations

The number of roots of the equation $|x^2-x-6|=x+2$ is

Number of distinct integer values of $a$ satisfying $2^{2a} - 3(2^{a+2}) + 25 = 0$ is:

Triangle sides: $x^{2}+x+1,\ 2x+1,\ x^{2}-1$. Largest angle?

If $a$ is a positive integer, then the number of values satisfying $ \displaystyle \int_{0}^{\pi/2} \left[ a^{2}\left(\frac{\cos 3x}{4}+\frac{3}{4}\cos x\right)+a\sin x - 20\cos x \right] dx \le -\frac{a^{2}}{3} $ is

Find least integer $k$ such that $(k-2)x^2 + k + 8x + 4 > 0$ for all $x\in\mathbb{R}$.

If $x^{2} + 2ax + 10 - 3a > 0$ for all x ∈ R, then

If $\alpha$ and $\beta$ are the roots of the equation $2x^{2}+ 2px + p^{2} = 0$, where $p$ is a non-zero real number, and $\alpha^{4}$ and $\beta^{4}$ are the roots of $x^{2} - rx + s = 0$, then the roots of $2x^{2} - 4p^{2}x + 4p^{4} - 2r = 0$ are:

The value of k for which the equation $(k-2)x^{2}+8x+k+4=0$ has both real, distinct and negative roots is