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The quadratic equation $x^2-5x+t =3x$ has only positive integer roots. Find the average of all distinct possible values of $t$.

 Dec 10, 2023
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The quadratic equation x2 – 5x + t = 3x has only positive integer roots. Find the average of all distinct possible values of t.  

 

Subtract 3x from both sides        x2 – 8x + t = 0    

 

Note that the resulting   

equation will factor.                     (x – 1)(x – 7) = 0   gives us t =   7  

                                                   (x – 2)(x – 6) = 0   gives us t = 12  

                                                   (x – 3)(x – 5) = 0   gives us t = 15  

                                                   (x – 4)(x – 4) = 0   gives us t = 16  

 

                                            7 + 12 + 15 + 16          50

Average of possible t's  =  –––––––––––––––  =  ––––  =  12.5  

                                                       4                        4  

.

 Dec 11, 2023

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