+0  
 
-1
14
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avatar+819 

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.

 Aug 27, 2023
 #1
avatar+1306 
+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 \(\frac{4}{10}=\frac{2}{5}.\)

 Aug 27, 2023
 #2
avatar+128826 
+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

 

cool cool cool

 Aug 27, 2023

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