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

If both the roots of the quadratic equation $x^2 - 2kx + k^2 + k - 5 = 0$ are less than $5$,then $k$ lies in the interval:

$f(x)=ax^2-bx-a$ is a quadratic expression. If $K$ is the least real number such that $f(x) \leq K, \forall x \in R$,then

The range of $a$ for which the roots of $x^2 - 2x - a^2 + 1 = 0$ lie between the roots (exclusive) of the equation $x^2 - 2(a + 1)x + a(a - 1) = 0$ is:

If $y = f(x) = ax^2 + 2bx + c = 0$ has imaginary roots and $4a + 4b + c < 0$,then :-

If the graph of $y = ax^2 + bx + c$ is as follows,where $\Delta ABC$ is a right-angled isosceles triangle with hypotenuse $AC = 4\sqrt{2} \text{ units}$,then the minimum value of $ax^2 + bx + c$ is: