Question: For what values of $m$ does $x^2−4x−m=0$ have no real solutions while $x^2−9x+m^2=0$ has at least one real solution?

Answer: 
**Hint:** Have a look at the discriminant $D=b^2-4ac$ for $f(x)=ax^2+bx+c$

If $D<0$ the equation $f(x)=0$ has no real solution. If $D\geq 0$ the equation $f(x)=0$ has at least one real solution.
take the first equation 
$$f(x)=x^2-4x-m$$
$$f'(x)=2x-4=0\rightarrow x=2$$
for real solution the $f(2)\geqslant 0$

and for no real solution if $f(2)<0$

For the second equation 

for real solution the $f(4.5)\geqslant 0$

and for no real solution if $f(4.5)<0$