Solution assignment 10 Quadratic inequalities

For which value(s) of $a$ do the parabola:

$P: y=x^2-1$

and the line:

$l: y=x+a$

have no points in common?

First we calculate the intersection points of $P$ and $l$ . Thus we have to solve:

$x^2-1=x+a$

The number of intersection points has to be $0$ and thus the discriminant of the equation:

$x^2-x-(1+a)=0$

has to be less than $0$ , thus if:

$1+4(1+a)<0$

$a<-\displaystyle\frac{5}{4}$