I choose a random integer $n$ between $1$ and $10$ inclusive. What is the probability that for the $n$ I chose, there exist no real solutions to the equation $x(x+5) = -n$? Express your answer as a common fraction.
+128407 x ( x + 5) = -n
x^2 + 5x + n = 0
For us to have a real solution here, the discriminant must be equal or greater to 0
So
5^2 - 4(1) n ≥ 0
25 - 4n ≥ 0
Note that this is true when n is an integer from 1 to 6 inclusive
So....we will have no real solutions when n= 7,8,9 or 10
So
P(no real solutions) = 4/10 = 2/5
