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Find all positive integer values of $c$ such that the equation $x^2-7x+c=0$ only has roots that are real and rational. Express them in decreasing order, separated by commas.

Guest Jan 23, 2018
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x^2  - 7x + c  =  0

 

If tthe roots are real then  this  must be true

 

(-7)^2  - 4c  > 0

 

49  - 4c  >  0

 

49 > 4c

 

49/4  > c

 

The positive integer  values that make this true  are   c = 1 - 12,  inclusive

 

But....for  rational solutions

 

49 - 4c    must be a perfect square

 

So...the integer values that satisfy this are

 

c  = 12, 10, 6

 

 

cool cool cool

CPhill  Jan 23, 2018

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