The number of real solutions of the equation $|x^2 + 4x + 3| + 2x + 5 = 0$ is:

If $\alpha, \beta$ are the roots of $ax^{2}+bx+c=0$ $(a \neq 0)$ and $\alpha+h, \beta+h$ are the roots of $px^{2}+qx+r=0$ $(p \neq 0),$ then the ratio of the squares of their discriminants is

The solution to the equation $4^{(x^2 + 2)} - 9 \cdot 2^{(x^2 + 2)} + 8 = 0$ is:

Find the complete set of values of $k$ for which the equation $4^x - (k + 2)2^x + 2k = 0$ has exactly one positive root.

Let $\alpha \neq \beta$ satisfy $\alpha^2+1=6 \alpha$ and $\beta^2+1=6 \beta$. Then,the quadratic equation whose roots are $\frac{\alpha}{\alpha+1}$ and $\frac{\beta}{\beta+1}$ is

If $f(x)$ is a quadratic expression such that $f(1) + f(2) = 0$,and $-1$ is a root of $f(x) = 0$,then the other root of $f(x) = 0$ is