The number of values of $k$ for which the equation $x^2 - 3x + k = 0$ has two real and distinct roots lying in the interval $(0, 1)$ is

The smallest value of $k$,for which both the roots of the equation $x^2-8kx+16(k^2-k+1)=0$ are real,distinct and have values at least $4$,is

If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

If non-zero real numbers $b$ and $c$ are such that $\min \,f(x) > \max \,g(x)$,where $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$ for $x \in R$; then $\left| \frac{c}{b} \right|$ lies in the interval

If $b > a$,then the equation $(x - a)(x - b) = 1$ has