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Find the sum of all possible positive integer values of b such that the quadratic equation 2x^2 + 15x + b = 0 has rational roots.

 Feb 27, 2022
 #1
avatar
+1

Rational roots means the discriminatn of the quadratic formula is > 0

   15^2 - 4(2)(b) > 0

       225/ 8 > b 

        28.125 > b            sum 1 - 28 for the answer                 

 Feb 28, 2022
 #2
avatar+36416 
+2

Rational roots means the discriminant of the quadratic formula is >= 0

   15^2 - 4(2)(b) >= 0

       225/ 8 >= b 

        28.125 >= b         

As alan correctly points out below ..... this will give some irrational numbers

The values between 1 and 28 that give RATIONAL numbers would be

0  1  4  9  16  and 25    I believe .... add 'em up

 

 

  ~ EP

ElectricPavlov  Feb 28, 2022
edited by ElectricPavlov  Feb 28, 2022
edited by ElectricPavlov  Feb 28, 2022
 #3
avatar+32957 
+2

Hmm.  For the roots of the equation to be rational it requires more than just that the discriminant is greater than or equal to zero.  It also requires that the square root of the discriminant is not irrational.

 Feb 28, 2022
edited by Alan  Feb 28, 2022
 #4
avatar+36416 
+1

Thanx , Alan.....  I frequently  seem to transpose  irrational and  complex  numbers.....

 

   I edited my answer above to be correct ...I think !

ElectricPavlov  Feb 28, 2022
 #5
avatar+32957 
+2

This should make it clearer:

Alan  Feb 28, 2022
 #6
avatar+36416 
+1

Ahhh.... I get it now

sqrt (225-8b) >0   and a perfect square  (to be RATIONAL)

But this needs to be a perfect square

  so b = 7  13  18   22   25    27  28                     Thanx, Alan ! 

ElectricPavlov  Feb 28, 2022

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