The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in the interval $[0, 1]$.

Let $a, b, c$ be real numbers such that $a+b+c < 0$ and the quadratic equation $a x^{2}+b x+c=0$ has imaginary roots. Then:

The maximum and minimum values of $\frac{x^2 + x + 1}{x^2 - x + 1}$ are respectively ...........

Let $f(x) = (1 + b^2)x^2 + 2bx + 1$ and $m(b)$ be the minimum value of $f(x)$. If $b$ can take any real value,what is the range of $m(b)$?

Consider the quadratic equation $(c - 5)x^2 - 2cx + (c - 4) = 0$,where $c \ne 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and the other root lies in the interval $(2, 3)$. Then the number of elements in $S$ is

For what condition is the value of the quadratic expression $x^2 + 2bx + c$ always positive?