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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.

 Dec 24, 2020
 #1
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x ( x + 5) =   -n

 

x^2  + 5x  +  n    =  0

 

For us to  have a real solution here, the discriminant  must  be  equal or  greater to  0

 

So

 

5^2 - 4(1) n   ≥  0

 

25  - 4n  ≥  0

 

Note that  this is true when  n  is an  integer from  1 to 6  inclusive

 

So....we will have  no  real solutions when n=  7,8,9 or 10

 

So

 

P(no real solutions)  =  4/10   = 2/5

 

 

cool cool cool

 Dec 24, 2020

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