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For how many integer values of a does the equation x^2 +ax + 5a have integer solutions for x?

 May 2, 2019
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\(x^2+ax+5a = (x-i_1)(x-i_2)=x^2 - (i_1+i_2)x+i_1i_2 \\ 5a=i_1 i_2\\ i_1+i_2=a\\ 5a = i_1(a-i_1)\\ i_1^2-i_1a+5a=0\\ i_1 = \dfrac{-a\pm\sqrt{a^2-20a}}{2}\\ \sqrt{a^2-20a} \in \mathbb{N}+\{0\}\)

 

\(a=20\text{ and }a=36 \text{ seem to be the only solutions}\)

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 May 2, 2019

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