Question

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

Answer 1

The quadratic equation $x^2-x+m^2=0$ has no real roots if its discriminant is less than $0$ i.e. $D=b^2-4ac<0$

In the given equation $a=1,b=-1$ and $c=m^2$

$⟹1-4m^2<0$

$⟹1<4m^2$

$\Rightarrow m^2>\frac{1}{4}$.

$\Rightarrow m>\frac{1}{2}$ or $m<-\frac{1}{2}$

$\Rightarrow m\in (-\infty ,-\frac{1}{2})\cup (\frac{1}{2},\infty )$