+230 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.
+128460 x ( x + 5) = -n
x^2 + 5x + n = 0
To have no real solutions.....the discriminant must be < 0
So
5^2 - 4 (1) n < 0
25 - 4n < 0
25 < 4n
4n > 25
n > 25/4
n > 6.25
So n = 7, 8 , 9 ,10 provide no real solutions.....so 4/10= 2/5 of a chance that the n you select gives no real solutions
