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

The value of $k$ for which the equation $x^2 - 3x + k = 0$ has at least one real root in $[0, 1]$ is

The set of values of $a$ for which the inequality $x^2 - (a + 2)x - (a + 3) < 0$ is satisfied by at least one positive real $x$ is:

Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.
Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.

If the roots of ${x^2} + x + a = 0$ exceed $a$,then

If both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$,then $m$ lies in the interval.