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.
The values that lead to non-real solutions are 9 and 10, so the probability is 2/10 = 1/5.
+118609 x(x+5) = -n
\(x(x+5) = -n\\ x^2-5x+n=0\\ x=\frac{5\pm\sqrt{25-4n}}{2}\\ \)
this will NO real solutions if the number under the square root is negative.
\(25-4n<0\\ 25<4n\\ 4n>25\\ n>6.25\)
So there will be no real solution if n = 7,8,9 or 10
that is 4/10 = 2/5 probability
