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For what positive integers $c$, with $c < 100$, does the following quadratic have rational roots? \[ 3x^2 + 20x + c \]

 Jun 1, 2023
 #1
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For a quadratic to have rational roots, its discriminant must be a perfect square. The discriminant of the given quadratic is 400 - 12c. For this to be a perfect square, we must have 400 - 12c = k^2 for some positive integer k. Solving, we get c = 400/12 - k^2. Since c < 100, we see that k must be 1, 2, 3, or 4. This gives us the four possible values of c: 25, 20, 15, and 12.

(Note: The quadratic may have other roots, such as irrational roots, that are not integers. We are only concerned with whether or not it has rational roots.)

 Jun 1, 2023
 #2
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It says that it is wrong

 Jun 1, 2023
 #3
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the other user forgot to divide by 12 under the k^2

C = (400-k^2) / 12

 

k can = 16 which makes c = 12

k can = 14 which makes c = 17 

k can = 10 which makes c = 25

k can = 8 which makes c = 28

k can = 4 which makes c = 32

k can = 2 which makes c = 33

 Jun 1, 2023

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