Fox integral

Algebra Level 4

If R R is a positive integer such that R < 26 R<26 , what is the total number of integral solutions for the equation x 2 + 10 x + R = 0 x^{2} + 10x + R = 0 ?


The answer is 9.

Ramiel To-ong by Ramiel To-ong , 37, Philippines

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1 solution

we must look at the discriminant δ = 100 − 4 R \delta = 100 - 4R of the equation x 2 + 10 x + R = 0 , 0 < R < 26 x^2 + 10x + R = 0, 0<R<26 . δ \delta has to be a perfect square for that the equation x 2 + 10 x + R = 0 x^2 + 10x + R = 0 has integers solutions ⇒ \Rightarrow R = 9,16,21,24 and 25. For R =9, 16, 21 and 24 the equation has 2 integers solutions and for R =25 the equation has one integer solution.

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