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.
The first equation has no real solution provided $$(-4)^2-4(-m)=4(4+m)\lt 0,$$ while the second has at least one real solution provided $$(-9)^2-4m^2=3^2-(2m)^2\ge 0.$$

Both conditions are satisfied when both inequalities hold *simultaneously.* Can you continue now?