+817 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.
+1306 Hi there!
There are 4 integers for which there is no real solution. They are: 7, 8, 9, and 10. Therefore, the probability is \(\frac{4}{10}=\frac{2}{5}.\)
+129847 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
