I choose a random integer n between 1 and 10 inclusive. What is the probability that for the I chose, there exist no real solutions to the equation x(x + 2) = -n? Express your answer as a common fraction.
Rewrite as
x(x + 2) + n = 0
x^2 + 2x + n = 0
If we have no real solutions, the discriminant is < 0 ....so.....
2^2 - 4n < 0
4 - 4n < 0
4 < 4n
n > 1
So... no real solutions exist when n = [ 2, 10 ]
So 9 /10 proobability of no real solutions
