Question 697462
Call the smaller number x, so the larger integer is x + 1. The difference of their cubes is 631, so the equation is:<br>

{{{(x+1)^3 - x^3 = 631}}}, expanding: {{{(x+1)^3 = x^3 + 3*x^2 + 3*x + 1}}}:
{{{x^3 + 3*x^2 + 3*x + 1 - x^3 = 631}}}
{{{3*x^2 + 3*x + 1 = 631}}}, now we have a quadratic equation that can be solved for x:
{{{3*x^2 + 3*x - 630 = 0}}}, divide both sides by 3 to simplify:
{{{x^2 + x - 210 = 0}}}, this can be factored as<br>

(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:<br>

{{{15^3 - 14^3 = 3375 - 2744 = 631}}}