SOLUTION: What two consecutive integers have cubes that differ by 631?

Question 697462: What two consecutive integers have cubes that differ by 631?
Found 2 solutions by Alan3354, Positive_EV:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
What two consecutive integers have cubes that differ by 631?
--------
5 & 6

Answer by Positive_EV(69) (Show Source): You can put this solution on YOUR website!
Call the smaller number x, so the larger integer is x + 1. The difference of their cubes is 631, so the equation is:

, expanding: :

, now we have a quadratic equation that can be solved for x:
, divide both sides by 3 to simplify:
, this can be factored as

(x+15)(x-14) = 0. So, x can be 14 or -15. Since the problem doesn't specify if the integers are positive or negative, both solutions are fine: 14 and 15 or -14 and -15 are both acceptable. Check:

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