For what values of $m$ does the equation $x^{2} - x + m = 0$ possess no real roots?

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Solution

The quadratic equation $x^{2} - x + m = 0$ has no real roots if its discriminant is less than $0$ i.e. $D < 0$
$\Rightarrow b^{2} - 4 a c < 0$
$1 - 4 m < 0$
$1 < 4 m$
$\Rightarrow m > \frac{1}{4}$ .

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