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 $x^2 - 8x + n = 0$? Express your answer as a common fraction.

maximum Aug 27, 2023

#1**+2 **

Hi there!

There are 4 integers for which there is no real solution. They are: 7, 8, 9, and 10. Therefore, the probability is \(\frac{4}{10}=\frac{2}{5}.\)

dolphinia Aug 27, 2023

#2**+1 **

x^2 - 8x + n = 0

If the discriminant is < 0, then we have no real solutions....so

(-8)^2 - 4n < 0

64 < 4n

64/4 < n

n > 16

Any n from 1 to 10 inclusive will work !!!!!

We can easiy see this....the disriminant will be smallest when n= 10 → (8)^2 - 4(10) = 24 but will still be > 0

CPhill Aug 27, 2023