For what values of $m$ does the equation $x^2 - x + m = 0$ possess no real roots?
Given quadratic equation: $\;$ $x^2 - x + m = 0$ $\;\;\; \cdots \; (1)$
Comparing equation $(1)$ with the standard quadratic equation $\;$ $Ax^2 + Bx + C = 0$ $\;$ gives
$A = 1, \; B = -1, \; C = m$
Discriminant of equation $(1)$ is
$\Delta = B^2 - 4AC = \left(-1\right)^2 - 4 \times 1 \times m = 1 - 4m$
For no real roots, $\;$ $\Delta < 0$
i.e. $\;$ $1 - 4m < 0$
i.e. $\;$ $1 < 4m$
i.e. $\;$ $\dfrac{1}{4} < m$ $\;\;$ i.e. $\;\;$ $m > \dfrac{1}{4}$
$\therefore \;$ The given quadratic equation has no real roots $\;$ $\forall \; m \in \left(\dfrac{1}{4}, + \infty\right)$